Cellular Automata Tutorial.docx

Cellular Automata Tutorial.docx

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CellularAutomataTutorial

CellularAutomataTutorial

1.Introduction

Fromthetheoreticalpointofview,CellularAutomata（CA）wereintroducedinthelate1940'sbyJohnvonNeumann（vonNeumann,1966;Toffoli,1987）andStanislawUlam.Fromthemorepracticalpointofviewitwasmorelessinthelate1960'swhenJohnHortonConwaydevelopedtheGameofLife（Gardner,1970;Dewdney,1989;Dewdney,1990）.

CA'sarediscretedynamicalsystemsandareoftendescribedasacounterparttopartialdifferentialequations,whichhavethecapabilitytodescribecontinuousdynamicalsystems.Themeaningofdiscreteis,thatspace,timeandpropertiesoftheautomatoncanhaveonlyafinite,countablenumberofstates.Thebasicideaisnottotrytodescribeacomplexsystemfrom"above"-todescribeitusingdifficultequations,butsimulatingthissystembyinteractionofcellsfollowingeasyrules.Inotherwords:

Nottodescribeacomplexsystemwithcomplexequations,butletthecomplexityemergebyinteractionofsimpleindividualsfollowingsimplerules.

HencetheessentialpropertiesofaCAare

∙aregularn-dimensionallattice（nisinmostcasesofoneortwodimensions）,whereeachcellofthislatticehasadiscretestate,

∙adynamicalbehaviour,describedbysocalledrules.Theserulesdescribethestateofacellforthenexttimestep,dependingonthestatesofthecellsintheneighbourhoodofthecell.

Thefirstsystemextensivelycalculatedoncomputersis-asmentionedabove-theGameofLife.Thisgamebecamethatpopular,thatascientificmagazinepublishedregularlyarticlesaboutthe"behaviour"ofthisgame.Contestswereorganizedtoprovecertainproblems.Inthelate1980'stheinterestonCA'sarisedagain,aspowerfulcomputersbecamewidelyavailable.Todayasetofacceptedapplicationsinsimulationofdynamicalsystemsareavailable.

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2.AGlanceatDynamicalSystemsandChaosor"AnewSciencenamedComplexity"

NatureandNature'sLawlayhidinnight,

Godsaid,letNewtonbe!

andallwaslight!

ThisproposedepitaphfromAlexanderPopeforIsaacNewtonsgraveshowsthementalattitudeofthe19thcentury.Sincethebeginningofthe20thcenturythescientificcommunityhadtodismiss（withaheavyheart）theideaofthecapabilitytocalculatethefuturestateofaphysicalsystemexactlyifthecurrentstateofthesystemisknownwithahighprecision.

ThisideaiscalledinamorephilosophicalterminologytheLaplacianDemon,referingtoanidealdemon,knowingpreciselythecurrentstateofasystem,mustbeabletoforeseetheexactdevelopmentofthissystem.Inthebackofthemindwasthebelieve,thatsmallerrorsinanalysationofthecurrentsystemwillcausesmallerrorsinextrapolationoffuturedevelopmentofthesystem.）

Iwanttopointoutthattheseideashadtofallregardingafewselectedmilestones,thatshowthenecessityforthechangeoftheparadigm:

∙HenriPoincareandthe"solution"ofthe3-bodyproblem

∙EdwardLorenzproblemofweatherforecast

∙WernerHeisenbergandtheuncertaintyrelation

OscarIItheSwedishking,whowasveryinterestedinmathematicsfoundedapricewonbyPoincareproofing,thatthereisnoclosedformmathematicalsolutionforthisproblem.（Besides:

ThearticleofPoincarewasthatchaotic,thataprintedandpartiallysoldcirculationhastobereprinted,asthelastchangesfromPoincarearrivedthatlate.ThisreprinthastobepaidbyPoincare,sothatthesumhereceivedfromOscarIIwasmorethanspent.

ThecontactofLorenztochaoticsystemsoccurredindoingweatherforecasts.Totellitinfewwords:

tryingdifferentmodels（andtryingtospeedupthecalculations）Lorenzreducedtheprecisionofoneparameter,thinking,thatthiswillleadinlittlereducedaccuracyintheresultofthecalculation.Theastonishingoutcomewasacompletelydifferentresult!

Aconsequencebothmendrawis,thatcomplexdynamicalsystemsoftenshowhugeeffectsonsmallchangesinthestartingconditions.

WernerHeisenbergsupplementsthefinding,thatitisimpossibletomeasurethepositionandspeedofaparticlewithhighprecisionatthesametime.Thiscognitionisoneofthebasisbricksofthequantumtheory.PuttingtheresultsofLorenzandPoincaretogetherwiththeuncertaintyrelationHeisenberg'sonecanseeclearly,thattheLaplacianDemonhaslostit'srighttoexist.Thiswasashockformostofthescientists.Newattemptstotheunderstandingofcomplexdynamicalsystemswerenecessary.WiththeworkofBenoitMandelbrotthenewscienceoffractal（broken）dimensionsandchaosstarted,recentlyamoreunifiedapproachcalledthescienceofcomplexityemerged.

Tryingtogiveadefinitionofcomplexityshowsthedifficultsituationthisnewbranchstucksin.ThephysicistSethLloydfromtheMITfoundthatmorethan30differentdefinitionsofComplexityexist.

NeverthelessadefinitionfromChristopherLangtonfromSantaFeInstituteseemstobeessential,describingComplexityasthesciencetryingtodescribethestatesontheedgeofchaos.Inotherwords,heassumes,thatorderemergesfromsystemsontheedgeofchaos.Inmoregeneraltermsonecanposethequestion:

"Howarisesorderfromchaos?

".

Astherearealotofdifferentstreamsinchaosandcomplexityresearch,itisnotthespaceheretodiscusstheseinterestingdevelopmentsindetail.Ijustfeltthenecessitytogetintotouchwiththeseitems.FormoreseriousintroductionsIcanrecommendMcCauley（1993）andOtt（1994）.

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3.BuildingCellularAutomata

3.1TheCell

ThebasicelementofaCAisthecell.Acellisakindofamemoryelementandstores-tosayitwitheasywords-states.Inthesimplestcase,eachcellcanhavethebinarystates1or0.Inmorecomplexsimulationthecellscanhavemoredifferentstates.（Itiseventhinkable,thateachcellhasmorethanonepropertyorattributeandeachofthesepropertiesorattributescanhavetwoormorestates.）

3.2TheLattice

Thesecellsarearrangedinaspatialweb-alattice.Thesimplestoneistheonedimensional"lattice",meaningthatallcellsarearrangedinalinelikeastringofperls.ThemostcommonCA'sarebuiltinoneortwodimensions.WhereastheonedimensionalCAhasthebigadvantage,thatitisveryeasytovisualize.Thestatesofonetimestepareplottedinonedimension,andthedynamicdevelopmentcanbeshownintheseconddimension.AflatplotofaonedimensionalCAhenceshowsthestatesfromtimestep0totimestepn.ConsideratwodimensionalCA:

atwodimensionalplotcanevidentlyshowonlythestateofonetimestep.Sovisualizingthedynamicofa2DCAisbythatreasonmoredifficult.

Bythatreasonsandbecause1DCA'saregenerallymoreeasytohandle（thereisamuchsmallersetofpossiblerules,comparedto2DCA's,aswillbeexplainedinthenextsection）firstofalltheonedimensionalCA'shavebeenexploitedbetheoreticians.Mosttheoreticalpapersavailabledealwithpropertiesof1DCA's.

3.3Neighbourhoods

Uptonow,thesecellsarrangedinalatticerepresentastaticstate.Tointroducedynamicintothesystem,wehavetoaddrules.The"job"oftheserulesistodefinethestateofthecellsforthenexttimestep.Incellularautomataaruledefinesthestateofacellindependenceoftheneighbourhoodofthecell.

Differentdefinitionofneighbourhoodsarepossible.Consideringatwodimensionallatticethefollowingdefinitionsarecommon:

vonNeumannNeighbourhood

fourcells,thecellaboveandbelow,rightandleftfromeachcellarecalledthevonNeumannneighbourhoodofthiscell.Theradiusofthisdefinitionis1,asonlythenextlayerisconsidered.

MooreNeighbourhood

theMooreneighbourhoodisanenlargementofthevonNeumannneighbourhoodcontainingthediagonalcellstoo.Inthiscase,theradiusr=1too.

ExtendedMooreNeighbourhood

equivalenttodescriptionofMooreneighbourhoodabove,butneighbourhoodreachesoverthedistanceofthenextadjacentcells.Hencether=2（orlarger）.

MargolusNeighbourhood

acompletelydifferentapproach:

considers2x2cellsofalatticeatonce.formoredetailstakealookattheApplications

vonNeumannNeighbourhood

MooreNeighbourhood

ExtendendMooreNeighbourhood

Theredcellisthecentercell,thebluecellsaretheneighbourhoodcells.Thestatesofthesecellsareusedtocalculatethenextstateofthe（red）centercellaccordingtothedefinedrule.

Asthenumberofacellsinalatticehastobefinite（bypracticalpurposes）oneproblemoccursconsideringtheproposedneighbourhoodsdescribedabove:

Whattodowithcellsatborders?

Theinfluencedependsonthesizeofthelattice.Togiveanexample:

Ina10x10latticeabout40%ofthecellsarebordercells,ina100x100latticeonlyabout4%ofthecellsareofthatkind.Anyway,thisproblemmustbesolved.Twosolutionsofthisproblemarecommon:

1.Oppositebordersofthelatticeare"stickedtogether".Aonedimensional"line"becomesfollowingthatwayacircle,atwodimensionallatticebecomesatorus.

2.thebordercellsaremirrored:

theconsequencearesymmetricborderproperties.

Themoreusualmethodisthepossibility1.

3.4ApplyingRules

Anexampleof"macroscopic"dynamicresultingfromlocalinteractionis"thewave"ina-saysoccer-stadium.Eachpersonreactsonlyonthe"state"ofhisneighbour（s）.Iftheystandup,hewillstanduptoo,andafterashortwhile,hesitsdownagain.Localinteractionleadstoglobaldynamic.Onecanarrangetherulesintwo（three）classes:

1.everygroupofstatesoftheneighbourhoodcellsisrelatedastateofthecorecell.E.g.consideraone-dimensionalCA:

arulecouldbe"011->x0x",whatmeansthatthecorecellbecomesa0