Cellular automata

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Figure1Demonstrationprogramofcellularautomata(3photos)

Tounderstandcellularautomata,lookatasimpleexample:findOnadrawingwithmanygrids,youcangetapattern(pattern)byblackingoutsomeofthegridswithapencil.Oneorseveralgridsinthefirstrowmaybeblackedout,andasimplecellularautomatonistodeterminesomesimpleruleanddrawnewpatternsfromthesecondrowdown.Specificallyforeachgridineachrow,observethecorrespondinggridinthepreviousrowandthesituationonbothsidesofthecorrespondinggrid,andthendeterminewhetherthethreegridsareblackedoutandhowtheblackandwhitegridsareadjacenttotheestablishedrules(forexample,Whenthethreegridsareblack,black,andwhitefromlefttoright,thegriddirectlybelowitiswhite,otherwiseitisblack),determinewhetherthecurrentgridispaintedblackorleftwhite.Thisisrepeated.Oneoragroupofsuchsimplerulesandsimpleinitialconditionsconstituteacellularautomaton.

Cellularautomatatheorymainlystudiesthetheoreticalmodelofsmallcomputersorcomponentsthatareconnectedinaneighborhoodconnectionmodeintolargercomputersorcomponentsthatworkinparallel.

AllthecellsintheNeumanncellspaceareonthenodesoftheintegergrid,andthenumberofcellsisinfinite.Itsatisfiesthefollowingconditions:eachcellisacertainMoorefiniteautomata;adoptsafive-neighborhoodconsistentconnectionmode(allcellshavethesameshapeofneighborhood);doesnottakeexternalinputanddoesnotoutputtotheoutside;anditisstatic(Theneighborhooddoesnotchangeovertime).Thegeneralcellspacedoesnotrequiretheseconditions,sotherearealsonon-deterministiccellspaces,Mire-typecellspaces,cellspaceswithinconsistentconnectionpatterns,cellspaceswithexternalinput,anddynamiccellspaces.

Thecheckerboardspaceisadirectextensionofthecellspace.Ithasaunifiedexternalinputassignedtoeachcell.Inotherwords,thecheckerboardspaceisacellspacecontrolledbyaprogram.Eachcellinthecheckerboardspacecanbeimaginedashavingafinitesetoflocaltransferfunctions.Therefore,thecheckerboardspacehasafinitesetofglobaltransferfunctions.Each"instruction"intheprogramselectstheglobaltransferfunctionusedinthetransferatthatmoment.

Mostcellularautomataproduceboringmonotonouspatterns,butsomeofthemarebeyondpeople’sexpectations.

Classification

(1)Thesimplestone-dimensionalcellularautomata

Thestatesetofthesimplestone-dimensionalcellularautomataistwoelements{0,1}.Theneighborisanareawitharadiusof1,thatis,thetwosquaresontheleftandrightofeachsquareareitsneighbors,sothateachsquarecellanditsneighborscanbeexpressedasfollows:

blackThesquareofisthecurrentcell,andthegraysquaresonbothsidesareitsneighbors.Sincethestatesethasonlytwostates{0,1},thatistosay,thesquarecanonlyhavetwocolorsofblackandwhite,thenanyonesquareplusitstwoneighbors,thestatecombinationofthesethreesquaresThereare8kindsintotal.

Thestatestheyindicateare:111,110,101,100,011,010,001,000.Thatistosay,foralltheone-dimensionalcellularautomatawhoseneighborradiusis1andthenumberofstatesis2,thereareonly8combinationsoftheirownstates.

(2)Rulesandnumbers

Therulesareconsideredbelow.Assumingthatthecurrentlyconsideredcellisci,hisstateattimetissi,t,anditstwoneighborsstatearesi-1,t,si+1,t,thenthestateofciatthenexttimeissi,t+1,theconversionruleisexpressedasafunction:

si,t+1=f(si-1,t,si,t,si+1,t)

,Si,t∈{0,1},foranyiandt

Becauseinoursimplestcellularautomaton,allpossiblecombinationsofeachcellanditsneighborstatesarelistedaboveThereare8types,soitsinputisoneofthe8combinationslistedabove.Theoutputrepresentsthestateofeachcellatthenextmoment,andthestatecanonlybe0or1,sotheoutputoftheruleiseither0,or1.Inthisway,anyruleisoneorasetofconversions,

Thenthissetofrulescorrespondstothecode:10100011,whichistoarrangethesquaresintheeightpositions.Wecanconvertthebinarycodeoftheoutputpartintoadecimalnumber:163,whichisthecodeofthecellularautomaton.Whenthenumberofstatesincreasesandtheradiusincreases,thisencodingmethodisnotpractical,andweneedtouseanothermethodtoencode.Considerthefollowingrule.Ifthereisarule:"Ifthereisonlyoneblacksquareamongthethreeinputsquares,thenthecurrentsquarewillbeblackatthenextmoment;iftherearetwoblacksquares,thenexttimewillbewhite,Iftherearethreesquares,thenextmomentisblack,ifthereare4squares,thenthenextmomentiswhite"canbeexpressedasthefollowingfunctiontable:

si,t+1=1,Ifsi-1,t+si,t+si+1,t=1

si,t+1=0,ifsi-1,t+si,t+si+1,t=2

si,t+1=1,ifsi-1,t+si,t+si+1,t=3

si,t+1=0,ifsi-1,t+si,t+si+1,t=0

wheresi,t∈{0,1},foranyiandt

Inthiscase,thereareonly4casesofinput,sothefollowingtablecanbeobtained:

Forthesamereason,wecanencodeitas:0101,whichis5indecimal.Obviously,thiscodingmethodisshorterthanthepreviousone,butthiscodingmethodcannotreflectallcellularautomata.

(3)Thedynamicbehaviorofthesimplestone-dimensionalcellularautomata

Fortheone-dimensionalcase,weassumethatallthesquaresaredistributedonastraightline,andthelengthofthestraightlineIsthewidthofouranimationarea,forexample,400,whichmeansthatthereare400squaresonthisstraightline.Weuseblackgridstorepresentthe1stategridsonthestraightline,andwhitegridstorepresentthe0stategrids.Thenanintermittenthorizontallineisadistributionofthecurrentstateofallcells.Thesesquareschangeovertimeandformdifferenthorizontallines.Weputthesetime-varyinglinestogetherlongitudinallytoformagridarea.Theverticalaxisrepresentsthepassageoftime(thedownwarddirectionispositive),andthehorizontalaxisrepresentsthestateofthecellularautomataatthecorrespondingmoment,andanimagecanbeobtained.Thisiswhattheaboveexampleprogramperforms.Changedifferentencodingparameters,andyouwillseeandobservetheirdynamicbehavior.

Inthecaseofthesimplestcellularautomata(thenumberofstatesis2,theradiusis1),thesecellularautomataaredividedintothreecategories.ObservethecellularautomataNo.224(longcode),somecellsappearfromtoptobottom,andthengraduallybecomeallwhite,thatistosay,afterafewtimesteps,allthecellularautomatabecomefixedstate0(Thatis,thewhitesquares),andneverchange.ThecellularautomataNo.132andNo.203havebecomeseveralverticallines.Don'tforgetthateachrowisastateofthecellularautomataatacertainmoment,soaverticallinecanbeformedintheverticaldirectiontoindicatethatthestateofthecellhasnotchangedonthetimeaxis.SoNo.132,No.203andNo.224areattractedtoafixedstate.

LookatCellularAutomataNo.208again,itisanumberofdiagonallines.Sinceourboundaryiscyclic,itcanbepredictedthatafterseveralperiodsofoperation,thecellularautomatawillreturntoitsoriginalstate,sosuchacellularautomataiscyclic.Thetimestepelapsedbetweentwoidenticalstatesisthecycleofthiscellularautomaton.LookingatthecellularautomataNo.150andNo.151,theyobviouslyhaveneitherafixedperiodnorapointthattheyareattractedto.Theyareinachaoticanddisorderedstate,whichwecallachaoticstate.Byrepeatedlyrunningthesimplestcellularautomataprogram,itisnotdifficulttofindthatall256typesofcellularautomatacanbeclassifiedintooneofthesethreecategories:fixedvalue,periodiccycle,andchaos.

Wecanguess,arethedynamicbehaviorsofallcellularautomataofthesethreetypes?Letusexpandthescopeofexplorationtoaslightlymorecomplicatedsituation.Weconsiderthatthenumberofstatesis2,andtheneighborradiusis2(thatis,eachcellhas4neighbors,twoontheleftandrightsides),whichisstillone-dimensional.Condition.Insuchacellularautomaton,inadditiontothethreecategoriesdescribedabove,wealsofoundanothertype.PleaselookatthetwocellsNo.20(accordingtotheshortcodingscheme)andNo.52(accordingtotheshortcodingscheme)Thedynamicoperatinggraphoftheautomatonissoweird,likeanupside-downvine.Thiskindofvineisacomplexstructure,itisneitherequivalenttocompletelyrandom,andthereisnosignofafixedcycle.Thiskindofcomplexstructureisexactlythetypeweareinterestedin,becauseitisneitherattractedtoafixedpointorperiodicstatetobecomerigid,nortooactivebecauseofrandomness;itnotonlyguaranteesacertainflowactivity,butalsoItcanalsoproducea"memory"structure.Theoperatingsituationisobviouslydifferentfromthethreecategoriesdescribedabove,sowecallitcomplex.Continuingtoruntheone-dimensionalcellularautomatawithvariousparameters,wefindthatalmostallthedynamicbehaviorsoftheone-dimensionalcellularautomatacanbedividedintothesefourcategories.

Basedontheabovediscussion,weclassifycellularautomataintofourcategories,whichare:

I.Fixedvaluetype:cellularautomatastaysinAfixedstate;

II,Periodictype:Cellularautomatacyclicallycirculatesbetweenseveralstates;

III,Chaostype:CellularautomataisinacompletedisorderInarandomstate,thereishardlyanylaw;

IV,complextype:Thecellularautomatamayproducecomplexstructuresintheprocessofoperation.Thisstructureisneithercompletelyrandomnorfixed.Cycleandstatus.

Weonlyintroducedtheone-dimensionalcellularautomataabove,andthetwo-dimensionalcellularautomataisnothingmorethanoneofthesefoursituations.Infact,letusthinkaboutwhattypeof"GameofLife"weintroducedearlier?OfcourseitshouldbetheIV.Onlycomplextypeswillbringuseternalnovelty.

Significance

Cellularautomatacannotonlyformallybestudiedasatheoreticalmodelofparallelcomputers,butalsoasalanguage(collectionofinputwordsacceptedbythemachine)recognizer.Alanguagerecognizedbyacertainrecognizermeansthattherecognizernotonlyacceptswordsinthelanguage,butalsorejectswordsthatdonotbelongtothelanguage.Whenthedimensionalityishigherthan1,speechrecognitionissometimesregardedaspatternrecognition.Forasuperpositionalautomaton,ifonlyoneletterisinputateachtimestep,afterallthewordsareinput,iftheinputandoutputcellentersaspeciallydesignedacceptancestate,itisconsideredtohaveacceptedtheword.Whenallthewordsofthelanguageareaccepted,itiscalledasuperimposedautomatalanguage.Similarly,checkerboardautomataandone-dimensionalcellularautomatacanalsobeusedaslanguagereceptors.

Theparallelcomputingmethodofcellularautomatacanrealizethedesignofsomeparallelcomputersandrecognizers.Cellularautomataisofgreatsignificancetothedesignmethodofintegratedcircuits.Large-scaleintegratedcircuitshaveobviousadvantagesintheformofcellarrays.Biologypromotestheoreticalresearchonautomata.Inturn,thedevelopmentofautomatatheoryprovidesamathematicalmodelandmethodforbiologicaldevelopment.Theresearchofcellularautomataiscloselyrelatedtotheresearchofformallanguage.Therecognitionabilityofvariouscellularautomataandtherelationshipbetweenthevariouslanguages​​thattheycanrecognizeandvariousformallanguages​​arestillunderdiscussion.Inaddition,thenatureofvarioustypesofcellularautomataandtherelationshipbetweenthemarealsotopicsofconcerntopeople