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Asynchronous cellular automata - special networks local slowdown produces global speedup

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Asynchronous cellular automata - special networks local slowdown produces global speedup
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Ardestani, Arash Khani
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ABSTRACT: Information processing in living tissues is dramatically different from what we see in common man-made computer. The data and processing is distributed into the activity of cells which communicate only with neighboring cells. There is no clock for the global synchronization of cellular activities. There is not even one bit of central memory for globally shared data. The communication network between cells is highly irregular and may change without changing the outcome of the computation. A simple network of automata is introduced and analyzed to represent a mathematical model of special group of cells in an imaginary tissue sample. The interaction between the cells, their communication method, and their level of intelligence is studied. Three different structures of this model are demonstrated. Later on a simplification in the cells' program and elimination of a beat keeping clock will lead to a finite state automata network that is surprisingly more powerful in achieving the overall network's goal than its previous generation which had the advantage of more complex programs and a beat keeping clock.
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Thesis (M.A.)--University of South Florida, 2009.
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by Arash Khani Ardestani.
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AsynchronousCellularAutomata-SpecialNetworks LocalSlowdownProducesGlobalSpeedup by ArashKhaniArdestani Athesissubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofArts DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:RichardStark,Ph.D. NatasaJonoska,Ph.D. MileKrajcevski,Ph.D. DateofApproval: March31,2009 Keywords:state,probability,serial,synchronous,halting c Copyright2009,ArashKhaniArdestani

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Acknowledgment IwouldliketothankDr.RichardStarkmymajorprofessorwhohelpedmedevelop theideasandexpandthemtothepointthatthispaperwasmadepossible.Otherspecial thanksgotomycommitteemembersDr.NatasaJonoskaandDr.MileKrajcevskifor takingthetimetoreviewthispaper.WithouttheircriticismandguidanceIwouldnot havebeenabletocompletethispaper.

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Contents Listofguresii Abstractiii Introduction1 2PartitioNprogramonanetwork3 2PartitioNprogramonasquarenetworkwithsynchronousactivity8 2PartitioNprogramonasquarenetworkwithserialactivity11 2PartitioNprogramonasquarenetworkwithasynchronousactivity16 Philosophyandconclusion32 References33 AppendixA34 i

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Listofgures 1Anexampleofa2PartitioNnetwork3 2A2PartitioNnetworkwithrandominitialvalues4 3A2PartitioNnetworkatastablestate4 4Basiclayoutofthe2PartitioNsquarenetwork6 5Allpossiblestatesofthe2PartitioNsquarenetwork6 6Generalstatetransition7 7One-stepstatetransition8 8Two-stepstatetransition8 9Statetransitionsofthesynchronoussquarenetwork9 10Behaviorofthe2PartitioNnetworkwithserialactivity11 11Globalstatetransitionsof2PartitioNnetworkwithserialactivity12 12Statetransitionsof2PartitioNnetworkwithasynchronousactivity17 13ExampleofstatetransitionandvaluesforP18 14Globalstatediagram-2PartitioNsquareasynchronousNetwork19 15Partialglobalstatediagram-2PartitioNsquareasynchronousNetwork20 16StatetransitionfromstateAtoallotherpossiblestates22 17Statetransitionwhenthereisastablecell23 18Example1ofhowtransitionprobabilitiesarecomputed23 19Example2ofhowtransitionprobabilitiesarecomputed24 20Transitionprobabilitieswithprobabilityofactivityasp28 21Transitionprobabilitiesforaninitialstatewithstablecell29 ii

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22Haltingtimevs.probabilityofactivity30 23Closeupofhaltingtimevs.probabilityofactivity31 iii

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AsynchronousCellularAutomata-SpecialNetworks LocalSlowdownProducesGlobalSpeedup ArashKhaniArdestani ABSTRACT Informationprocessinginlivingtissuesisdramaticallydierentfromwhatwesee incommonman-madecomputer.Thedataandprocessingisdistributedintotheactivity ofcellswhichcommunicateonlywithneighboringcells.Thereisnoclockfortheglobal synchronizationofcellularactivities.Thereisnotevenonebitofcentralmemoryfor globallyshareddata.Thecommunicationnetworkbetweencellsishighlyirregularand maychangewithoutchangingtheoutcomeofthecomputation.Asimplenetworkofautomataisintroducedandanalyzedtorepresentamathematicalmodelofspecialgroup ofcellsinanimaginarytissuesample.Theinteractionbetweenthecells,theircommunicationmethod,andtheirlevelofintelligenceisstudied.Threedierentstructuresofthis modelaredemonstrated.Lateronasimplicationinthecells'programandelimination ofabeatkeepingclockwillleadtoanitestateautomatanetworkthatissurprisingly morepowerfulinachievingtheoverallnetwork'sgoalthanitspreviousgenerationwhich hadtheadvantageofmorecomplexprogramsandabeatkeepingclock. iv

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1Introduction Let'sbeginwithsomebasicdenitionsandhistoricalinformation.Theword automaton isderivedfromtheGreekword automatos meaning"actingofone'sownwill". Automatonisgenerallyreferredtoasmachinesthatsimulatelivingorganisms'movementsandactions,withoutanyelectricalpartsorcomponents. Thehistoryofautomatacanbetracedbacktoatleast3000yearsago.Thereare manyevidencesofmachinesinancientGreece,includingthetoysandtoolsbuiltby Heron ,representingbasicscienticprinciples.AnotherexampleistheancientChinese mechanicalengineer YanShi alsoknownasthe Articer ,whodemonstratedalife-size humangureofhismechanicalhandiworktoKingMuofZhou. Throughoutthecenturiestheideaofdesigninglife-likemachinesandtoyscontinued byscientistsandenthusiastsallaroundtheworld.Themaingoalbehindeortshasbeen tocreateamachinethatcanselfoperate,basedonexistinglifeformsknowntoscientists. Althoughtheideabehindautomatawasstudiedandpracticedthroughoutthoseyears, butitwasmainlyfocusedonthehighlevelbehaviorofthesystem.Itwasnotuntil20 th centurywhenautomatawasviewedasanewtooltostudymuchlowerlevelsofbehavior withinthesmallestelementsofasystem.Thisisessentiallywhatisknownas cellular automata 1 wherewestudytheactivityofthebasicelementsofasystemtodetermine theoverallbehaviorofthatsystem. Inthe1940'sStanislawUlamdevelopedanewmethodtostudygrowthofcrystals, whilehiscolleagueJohnvonNeumannstudiedtheverybasicsofselfreplicatingmachines. 1 Aformaldenitionofcellularautomataisprovidedinchapter1. 1

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Thisisbelievedtobethebirthofwhatwerefertoasthecellularautomata.Sincevon Neumann'soriginalworkonself-replicatingmachines,muchworkhasbeendedicatedto studying,designing,andanalyzingnetworksofcellularautomata. Inthefollowingpaperwewillintroducedenitions,structures,andanalysisofsome simplenetworksofautomata.Thegoalofthispaperistoprovideageneraloverviewof aspecialkindofnetworkofautomataknownasasynchronousnetworkofautomata. Thispaperisintendedtodemonstratenaturali.e.biologicalcomputingmodeled onnetworksofautomata.Asweintroducedierentgenerationsofthisnetwork,the ideaistoshowthatthemorerelaxedtheconditionsandthesimplerthedesignofthe network,themorepowerfulitsresultingmodelwillbe.Ultimatelythegoalwouldbeto mathematicallyprovethatthemodelisinfactcapableofthisnaturalcomputingand thatitsirregularitywillleadtoamoreidealmodel. 2

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22PartitioNprogramonanetwork Letusbeginbyintroducingthenetworkandhowthe2PartitioNprogramworks. Thenetworkconsistsofnitelymanycellseachtakingoneofthetwovalues0or1.In thisnetworkeachcellisactuallyanite-stateautomatonthatisconnectedtonitely manyneighborsbyanon-directededge.The2PartitioNprogramisthendeployedon thisnetwork.The2PartitioNprogrammakeseachcelltobeabletochangeitsown valuebasedontheinputthatreceivesbyreadingitsneighbors'values.Ifthecellsees aneighborwiththesamevaluethanitsown,itwillchange.Otherwiseitwillstaythe same. InFigure1,whiteindicatesacell-valueof1andblackindicates0.0-0or1-1edges causeinstabilityinthenetwork.Unstablecellsaremarkedwith"u"andstablecellsare markedwith"s". Figure1: Anexampleof2PartitioNnetwork. Inthisnetworkcellsarerepresentedbyverticesandcell-cellcommunicationlinksare representedbytheedges.Initiallyallthecellshaverandomvalues. 3

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Figure2: A2PartitioNnetworkwithrandominitialvalues. Thecellsareactive,ornot,randomly.Whenactive,acell'svalueisre-computedaccordingtoitsprogram.Afterthousandsofchanges,thestill-livingcolonystabilizes| asshowninFigure3.Isthereaglobalmeaningtothisstability? Figure3: A2PartitioNnetworkatastablestate. Toanswerthisquestionweshallspendafairamountoftimeintheupcomingsections andinvestigatethenetwork'sbehavioringreatdetail;howevertheimmediateanswerwill begiveninthenextfewpageswhileweintroducethebasicdesignofthismodel. Amathematicalrepresentationof2PartitioNprogramwillbeasfollows:Itexistsona networkconsistingofaset C ofcells,andaset E C 2 ofcommunicationedgesbetween 4

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cells.Thecellsaredenedascopiesofanautomaton,andcell-cellcommunicationis denedbyan input function. C;E isassumedtobenite,tohavemorethanonecell, tobeconnectedbynon-directededges,andtohavenoself-edges. For2PartitioN,theautomatahavevalues Q = f 0 ; 1 g andavalue-transitionfunction value;input = 8 < : 1 )]TJ/F23 11.9552 Tf 11.956 0 Td [(value if input = value value otherwise. The input isdenedforamulti-set 2 M ofneighboringcell-values. input M = 8 < : 0if1 62 M; 1if0 62 M: Theinputdescribestheneighbors'values|if input =0thenatleastoneneighborhasa cell-valueof0,if input =1thenatleastoneneighborhasavalueof1.If input indicates thataneighborhasthesamevalueastheactivecell,then returns )]TJ/F23 11.9552 Tf 12.062 0 Td [(value foran activecell'snextvalue.Thismeansthataslongasthereareatleasttwoneighboring cellswiththesamevaluethenetworkisunstable;thereforetheactivecellswillcontinue tochangetheirvaluesuntilthenetworkgoesintoastateinwhichtherearenotwo neighboringcellsofthesamevalue;thusbecomingcompletelystable.Thenetworkis essentiallyabipartitegraph,sothisbehaviorisbasicallythe"globalstability"that wementionedinthepreviousquestion.Furtherinthispaperwewillre-introducethis conceptwhenwedenehalting. Asmallsubgraphofthe2PartitioNnetwork,called2PartitioNsquareNetworkwill bethefocusofthispaperfromthispointforward.Throughouteachsectionwewillvisit dierentarrangementsofthisnetworkandwhileanalyzingthenetworkbehaviorwewill compareeachcongurationtoanother.Intheendwewillrevealsomeveryinteresting andunexpectedresultwhichisprobablytheessenceofthispaper.Figure4showsthe basiclayoutofthesquareNetwork. 2 A multi-set islikeasetexceptthatelementsmayberepeated.As sets f 0 ; 1 ; 1 ; 1 g and f 0 ; 1 g are equal,butas multi-sets theyaredierent. 5

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Figure4: Basiclayoutofthe2PartitioNsquarenetwork. Inthisnetworkcellsarefourverticesinthefourcornersofasquare,withthesides ofthesquarerepresentingtheedgesbetweenthefourverticesofasimplegraph.Each vertexcantakeavalue0or1.Obviouslythereare4 2 =16distinctcongurations,since wehavefourcellsandforeachcellwehavetwochoicesi.e.0and1.Acompletelistof thesestatesisshownbelow: Figure5: Allpossiblestatesofthe2PartitioNsquarenetwork. Onemaynoticeasymmetrybetweendierentstatesofthemodel,butitisnecessary topointoutthatwhilethenetworkarchitectureishighlysymmetric,wemakeabsolutely nouseofthissymmetryanywherethroughoutthispaper.Inotherwordsweinsiston irregularcommunicationarchitecture.Thesquaredesignwassolelychosenduetoits simplicitywhichmakestheconcepteasiertodigest;howevertheentireideaisapplicable 6

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toanyirregularnetworksuchastheoneshowningure-1onpage3. 7

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Moreparticularlywerefertothesecongurationsas "networkstates" "globalstates" orwithamoregeneralterm states .Nextwedeneasetofsimplerulesthatgovernthe activityofthecellsinthenetwork: 1.twocellsareconsideredneighborsifthereisanedgebetweenthem. 2.eachcellcanreadthevalueofitsneighbors. 3.acellchangesitsvalueifatleastoneofitsneighborshasthesamevalueasthe cell,onlyifthecellisactive.Agraphicalrepresentationofthisnetworkisshownbelow: Figure6: Generalstatetransition. Inmathematicallanguagewecanrepresent:Let C bethesetofcellssuchthat C = f a;b;c;d g andlet E C 2 bethesetofedgesthatdenotethecommunicationedges betweencells.Thevaluesofcellsisdenedastheset Q = f 0 ; 1 g .The transition function isthesameaswasdenedearlier.Ifacellisan active cellthenitreadsthevaluesof itsneighborsandcomputesitsownvalueatthatgivenmoment.Otherwisethecellis inactive 8

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32PartitioNprogramonasquarenetworkwithsynchronousactivity Supposewerandomlypickoneofthestatesof2PartitioNnetwork.Ifallcells areactiveatthesametime,thenwecallthisnetworka synchronousnetwork .Inother words,acellisonlyactiveifallofitsneighborsareactiveatthegivenmoment.Let's lookatanexampleinmoredetails: Figure7: One-stepstatetransition. Thenetworkipsbetweentwostatesforever,sinceitloopsbetweentwostates.These statesareshowninthegureabove.Letuslookatanotherexample: 9

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Figure8: Two-stepstatetransition. Inthiscongurationtheactivityofeachcellislimitedtoonlyasetofnon-halting congurationssinceitdependsontheactivityofallitsneighbors.Suchbehaviormakes thiscongurationof2PartitioNarathernotinterestingnetworktostudy.Forexample givenarandominitialstate[0011]thenextstatewillbe[1100].Ifthenetworkisactivated again,itwillimmediatelygotonextstate[0011].Obviouslythisnetworkinsynchronous modewillipbackandforthbetweenmaximumtwostates,exceptforthetwohalting states[0101]and[1010].Thetwohaltingstatesneverchangetoanotherstate,regardless ofcellactivity.Thisfactisshowningure-10below.Thereasonforthisbehavioris thatinsynchronousmodeallthecellsareactiveatthesametime.Inotherwordsall ofthefourcellsreadthevaluesoftheirneighborsatthesametimeandchangetheir valuesbasedontherulesdenedunderthetransitionfunction.Completestatediagram isdemonstratedbelow: 10

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Figure9: Statetransitionsofthesynchronoussquarenetwork. 11

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Asseenabovethenumberofstepsfromonestatetoanotherislimitedtomaximum oftwostep.Thiscriteriaofthesynchronous2PartitioNnetworkmakesitsbehavior verysimpleandthusitsnextstatesverypredictable.Anotherinterestingfactaboutthe synchronousnetworkisthatitneverhalts,unlessitisalreadyinahaltingstate.What thismeansforthe2PartitionNsynchronousnetworkisthatthismodeofactivitycannot beusedtomodelthepartitionsofbipartitenetworks,sinceitislimitedtoonlyafew transitionstatesanditneverhalts. Anotherveryimportantaspectofthisnetworkisthatinorderforallcellstobecome activeatonce,thereisaninevitableneedforaclock.Whatwemeanbyaclockinthis caseisacentralprocessorprogramwhichisaccessiblebyallcellsandactsasabeat keeper.Itisobviousthattheneedforacentralclockinanynetworkwillbeinterpreted asadisadvantageintheeciencyofthecommunicationovertheentirenetwork.The lackofeciencyincommunicationmaynotbeapparentinasmallnetworksuchasthe 2PartitioNsquareserial;howeverinasignicantlylargernetworkitcertainlyintroduces arealchallenge.Asanexampleconsider10 12 skincellsofthehumanbodytryingtoread acentralclockevery0.050seconds.Onecanimaginehowthiswouldimpactthenetwork intermsoftheneedforadditionalcommunicationlinksbetweeneachcellandthecentral clock,nottomentiontheextratimeneededforeachcommunicationtransaction. 12

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42PartitioNprogramonasquarenetworkwithserialactivity Supposewearegiventhesame2PartitioNnetwork.Assumingwearegivenany initialstateofthenetwork,butweallowonlyonecell c 1 ;c 2 ;c 3 ;c 4 tobeactiveatanygiven time.Theresultingnetworkiswhatwegenerallyrefertoas2PartitioNserialnetwork. Thenetwork'sactivityisserialinthesensethatitscellstakeactiverolerandomlybut onecellatatime.Belowthereisademonstrationofhowthisnetworkmightbehave: Figure10: Behaviorofthe2PartitioNnetworkwithserialactivity. Itisworthytonotethatthecellsdonotneedtotakeactiveroleinanyspecic order.Amajordierencebetweentheserialnetworkandthesynchronousnetworkthat wasdiscussedinprevioussectionisthattheserialnetworkisa halting network,when thenetworkisbipartiteandcellactivityisrandom.Inotherwordsthisnetworkcango fromnon-haltinginitialstatetoahaltingstate.Thiscriteriagivesthisnetworkareal advantageoverthesynchronousnetwork.Thisisduetothefactthatahaltingnetwork ofthistypecansolvetheproblemofbipartitepartitioning.Belowthereisacomplete 13

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demonstrationofallstatetransitionsfrominitialstatestothenextstatesorhalting states: Figure11: Globalstatetransitionsof2PartitioNnetworkwithserialactivity. 14

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Wecancomputetheaveragehaltingtimeforthe2PartitioNsquarenetworkwith serialactivity.Todosoweneedtoformulatetheexpectedhaltingtimeinrelationto thetransitionprobabilityandtheprobabilityofactivityforagiveninitialstate: H =1+ X P H H theexpectedhaltingtimefromthestate forexample H 0011 thestatetowhichwehaveasinglesteptransitionfromstate H theexpectedhaltingtimeforthestate forexample H 0111 P theprobabilitythatstate willhaveonesteptransitionfromstate Note:Inthiscase P =1 = 4forall .Asanexampleletuswriteaformulaforcalculating theexpectedhaltingtimeforstate =0011: H 0011 =1+[1 = 4 H 1011 +1 = 4 H 0111 +1 = 4 H 0001 +1 = 4 H 0010 ] Tondtheaverageexpectedhaltingtimewewillfollowthesesimplesteps: Step1:Usingglobalstatetransitiondiagramndtheexpectedhaltingtimeforeach initialstateintermsofother Step2:Writetheequationforeachexpectedhaltingtimeintermsofeachinitialstate andotherstates Step3:Settheseequationsasasystemwith16equationsand14unknowns 3 andsolve itusingMaple Step4:Findtheaverageoftheexpectedhaltingtimes Letusbeginwithstep1: H 0011 =1+[1 = 4 H 1011 +1 = 4 H 0111 +1 = 4 H 0001 +1 = 4 H 0010 ] H 1011 =1+[1 = 4 H 0011 +1 = 4 H 1010 +1 = 4 H 1001 +1 = 4 H 1011 ] H 1111 =1+[1 = 4 H 0111 +1 = 4 H 1011 +1 = 4 H 1101 +1 = 4 H 1110 ] H 1001 =0 3 Thereareonly14unknownsbecause H 0110 and H 1001 arezero 15

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H 1010 =1+[1 = 4 H 0010 +1 = 4 H 1110 +1 = 4 H 1000 +1 = 4 H 1011 ] H 1101 =1+[1 = 4 H 0101 +1 = 4 H 1100 +1 = 4 H 1001 +1 = 4 H 1101 ] H 1110 =1+[1 = 4 H 1100 +1 = 4 H 1010 +1 = 4 H 0110 +1 = 4 H 1110 ] H 1000 =1+[1 = 4 H 1100 +1 = 4 H 1010 +1 = 4 H 1001 +1 = 4 H 1000 ] H 1100 =1+[1 = 4 H 0100 +1 = 4 H 1000 +1 = 4 H 1110 +1 = 4 H 1101 ] H 0111 =1+[1 = 4 H 0011 +1 = 4 H 0101 +1 = 4 H 0110 +1 = 4 H 0111 ] H 0101 =1+[1 = 4 H 1101 +1 = 4 H 0001 +1 = 4 H 0111 +1 = 4 H 0100 ] H 0110 =0 H 0100 =1+[1 = 4 H 1100 +1 = 4 H 0110 +1 = 4 H 0101 +1 = 4 H 0100 ] H 0001 =1+[1 = 4 H 1001 +1 = 4 H 0101 +1 = 4 H 0011 +1 = 4 H 0001 ] H 0000 =1+[1 = 4 H 1000 +1 = 4 H 0100 +1 = 4 H 0010 +1 = 4 H 0001 ] H 0010 =1+[1 = 4 H 1010 +1 = 4 H 0110 +1 = 4 H 0011 +1 = 4 H 0010 ] AftersolvingthesystemofequationsusingMaple,thefollowingvaluesareobtained: H 0011 =7 : 062499992 H 1011 =6 : 083333325 H 1111 =7 : 166666657 H 1001 =0 : 0 H 1010 =7 : 187499990 H 1101 =6 : 083333325 H 1110 =6 : 458333325 H 1000 =6 : 124999992 H 1100 =7 : 187499992 H 0111 =6 : 041666658 H 0101 =7 : 062499988 H 0110 =0 : 0 H 0100 =6 : 083333325 H 0001 =6 : 041666658 H 0000 =7 : 083333323 H 0010 =6 : 083333325. Theaverageexpectedhaltingtimeis6.553571419 Asmentionedearlierthebasicpropertyofaserialnetworkisapparentinthewaythecell activitytakesplace.Forthepurposeof2PartitioNsquarenetworkwithserialactivity, onlyoneofthefourcellscanbeactiveatatime.Thereisaninevitableneedfora mechanismtoensurethatnomorethanonecellcanbeactiveatagiventime.There aredierentmethodstoimplementthismechanism.Onemechanismwouldbeusinga 16

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clockjustliketheclockmentionedinthe2Partitionsquarenetworkwithsynchronous activity.Anothermechanismwouldbetouseatoken,similartothetechnologyusedin tokenringnetworks.Dependingonthesizeandstructureofthenetworkonemechanism maybepreferredoveranother,butthatsubjectisoutofthescopeofthisarticle.Inany eventenforcingserialactivityonthismodelrequiresasubstantialcomputationaleort. Theidealgoalwith2PartitioNnetwork-cellularnetworkofautomata-istobeableto designthecellsinsuchawaythatthememoryrequirementisminimaltononeandthat theyareprogrammedinthesimplestwaypossible.Inthenextsectionwewilldescribe aspecialkindof2PartitioNnetworkthatistheclosesttotheidealmodel. 17

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52PartitioNprogramonasquarenetworkwithasynchronousactivity Consideringthesame2PartitioNnetworkthatwehavebeenanalyzingsofar,but themajordierentthistimeisthatweallowanyofthecellstobeactiveatanygiventime. Eventhoughthischangeinthecells'activitymayseemminoratrst,yetdeeperanalysis ofthenetworkprovesthattheimpactonthenetwork'sbehaviorandmoreimportantly itshalt-abilityisabsolutelysignicant.Weintroducethisnetworkas2PartitioNsquare asynchronous.Thewordasynchronoushereismeanttodescribetherandomnessineach cell'sactivity.Atrstglanceitmightappearasiftheasynchronousnetworkdesign willbemorecomplexbutitturnsoutthattheasynchronousnetworkdoesnotneeda clockandeachcellisdesignedwithamuchsimplerprogram,comparedtothenetworks examinedintheprevioussections.Amajordierencebetweenthe2PartitioNnetwork withasynchronousactivityandthe2PartitioNnetworkwithsynchronousactivityisthat therstonewillalwayshalt,whilethelattermayormaynothalt. Itisinterestingtonoticethatthestatetransitionofthe2PartitioNsquarenetwork withserialandsynchronousactivityarebothspecialcasesofthe2PartitioNsquare networkwithAsynchronousactivity.Thisalsomeansthattheglobalstatediagrams ofthe2PatitioNsquarenetworkwithserialandsynchronousactivityaresubsetsofthe globalstatediagramforthe2PartitioNsquarenetworkwithasynchronousactivity.In 2PartitioNsquareasynchronousnetworkwerelaxtherulesthatgoverntheactivityof thecells;thusgivingthenetworkmoreexibilityandperhapsmorecomplexity. Beforegoingfurtherletusdemonstrateaguretofamiliarizeourselveswithhowthis networkbehaves. 18

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Figure12: statetransitionsof2PartitioNasynchronousnetwork. Obviouslythisprocesscontinuesuntilitreachesanyofthetwohaltingstates.As mentionedearlierthe2PartitioNsquarenetworkisahaltingnetwork.Itisimportantto mentionthat2Partitionmusthaltonbipartitegraphs. Theorem: 2PartitioNonsquarenetworkinasynchronousmodewillalwayshalt. Proof: Supposeweareatoneoftherandominitialstates.Thegreatestprobabilitythatthe nextstateisnotoneofthehaltingstatesisatmost P where P< 1.Supposewego tothenextnon-haltingstateandagaintheprobabilityofgoingtothenextnon-halting stateis P .Ifwecontinueinthisfashionfor n steps,thentheprobabilitythatwedo notreachahaltingstatein n stepsis P n .Ontheotherhandtheprobabilitythatthe networkhaltsin n stepsis 1 )]TJ/F23 11.9552 Tf 12.257 0 Td [(P n .As n goesto 1 then P n approaches0therefore 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(P n approaches1. lim n !1 P n =0 lim n !1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(P n =1 Thisshowsthat2PartitioNonsquarenetworkinasynchronousmodewillalwayshalt withprobability1.Tomakethispointclearerlet'sdemonstratethecasebelow: 19

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Figure13: ExampleofstatetransitionandvaluesforP. InthegureontheleftthePiscalculatedas P = )]TJ/F15 11.9552 Tf 12.259 0 Td [(1 = 8=7 = 8whereintheright gurewehave P =1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1=0. Figurebelowrepresentstheglobalstatediagramofthe2PartitioNsquareasynchronous network: 20

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Figure14: Globalstatediagram-2PartitioNsquareasynchronousNetwork. Todemonstratethismoreclearlylet'spresentthenextgure: 21

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Figure15: Partialglobalstatediagram-2PartitioNsquareasynchronousNetwork. Thegureabovecapturesapotionofthepreviousdiagramanddemonstrateshoweach initialstateisrelatedtootherstatesincludingthehaltingstates.Clearlynotallthe statesareshown,duetolackofspacetoincludeallarrowsandallstates,butthe guredeliversthepoint.Perhaps,thenextfewparagraphsexplainthedynamicsofthe networkbetter.Toobtainthenextstatesfromeachinitialstate,weconsiderallpossible 22

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combinationsofactivecellsa,b,c,andd,skippingtherepetitionsofcourse.Ingeneral wewillconsiderthefollowing16cases: Noneofthecellsareactive Case1:Nocellisactive Justsinglecellbeingactive Case2:Onlycell"a"isactive Case3:Onlycell"b"isactive Case4:.... ............ Allcombinationsoftwocellsbeingactiveatatime Case6:Cells"a"and"b"areactive Case7:Cells"a"and"c"areactive Case8:.... ............ ............ Allcombinationsofthreecellsbeingactiveatatime Case12:Cells"a"and"b"and"c"areactive Case13:Cells"a"and"b"and"d"areactive Allfourcellsactiveatatime Case16:Cells"a"and"b"and"c"and"d"areactive Mathematicallywearebasicallytakingallthesubsetsofthesetofcells C = f a;b;c;d g toobtainthesetofactivecells.Ineachcasethemembersofeachsubsetdenetheactive cellsforthatcase. Letusexamineoneofinitialrandomstatesandseehowitwouldgotoallother possiblestates: 23

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Figure16: StatetransitionfromstateAtoallotherpossiblestates. Nowthatwehaveseenthisconstructiononecaneasilyseetherelationbetweeneach initialstateandthenextpossiblestatesdemonstratedingure-16.Onecanquickly noticethatsomeoftheinitialrandomstatescanonlychangetoeightofthesixteen possiblenextstates.Thereasonbehindthiscongurationisthatinsomeoftherandom initialstatesoneofthecellsiswhatwecallastablecell.Astablecelliscellthatdoesnot haveanyneighborwiththesamevalueasitself;thusithasnoeectonthecomputation ofthenextstate.AccordingtothetransitionfunctiondenedinSection1,thestable celldoesnotneedtochangeitsvalueregardlessifitisactiveornot-aloneorinany combinationwithothercells-Letusdemonstratethisbehaviorinthefollowinggure, wherecell"b"isthestablecell: 24

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Figure17: Statetransitionwhenthereisafreecell. Sofarwehaveestablishedthefactthatthe2ParitioNsquarenetworkwithasynchronous activitywillalwayshalt.Additionallywehavedeterminedacompleteglobaltransition graph,whichshowstherelationbetweeneverypossiblerandominitialstateandallthe otherstates,includingthehaltingstates.Nowitwouldbeinterestingtoexamineeach randominitialstatetoseehowtheactivityofeachcellcanaectthenexttransition state.Tostudyandanalyzethisprocesswewilltakeadvantageofthenotionoftransition probability.Whatwemeanbytransitionprobabilityistheprobabilitythatagiven randominitialstatewouldgotothenextpossiblestate.Wewillstudytherelation betweentheprobabilityofactivityforeachcellinagivenrandominitialstateandthe transitionprobability.Letusbeginwiththefollowingexample: Figure18: Example1ofhowtransitionprobabilitiesarecomputed. 25

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. Foreachactivecellweassignthevaluepandforeachinactivecellweassignthevalue -p.Ifthereisastablecell,theprocessisthesame,butthetotalprobabilitieswill beaddedtogetherjustlikeinthefollowingexample: Figure19: Example2ofhowtransitionprobabilitiesarecomputed. Inexample1tocomputethetotalprobabilitiesgoingfrom1010to1110wehave: P 1010 1110 = )]TJ/F23 11.9552 Tf 11.956 0 Td [(p 3 p Inexample2tocomputethetotalprobabilitiesgoingfrom0111to0110wehave: P 0111 0110 = )]TJ/F23 11.9552 Tf 11.956 0 Td [(p 3 p + )]TJ/F23 11.9552 Tf 11.955 0 Td [(p 2 p 2 Bydesign,weleteachcellinthesquarenetworkwithasynchronousactivitytobeactive orinactiveatanygiventime.Thereforetheprobabilityofacellbeingactiveisp=1/2. Similarlytheprobabilityofacellbeinginactiveisalso1 = 2i.e.1-p=1/2.There arefourcellsinthesquarenetwork,sothetotalprobabilityforgoingfromonestateto anotheriscalculatedas1 = 16.Ofcourseinsomeofthestateswherewehavestablecells, theprobabilitywouldbe2 P =1 = 8. Asseeningure-17eachrandominitialstatewillendonatleastoneofthehalting states.ThisisvisualconrmationoftheproofofTheorem-1.Thefactthatthisnetwork isahaltingnetworkbringsupafewinterestingquestions: 26

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Whatistheexpectedhaltingtimeofthe2PartitioNasynchronousnetwork? Doesthechoiceofinitialstateaecttheaverageexpectedhaltingtime? ToanswerthesequestionwewillusethesamemethodasweusedinSection3pages13-14: H =1+ X P H H theexpectedhaltingtimefromthestate forexample H 0011 thestatefromwhichwehaveasinglesteptransitionfromstate H theexpectedhaltingtimeforthestate forexample H 0111 P theprobabilitythatstate willbeobtainedinonesteptransitionfromstate Itisimportanttonotethatinthecurrentcase P =1 = 16forall .Soifwewereto writeaformulaforcalculatingtheexpectedhaltingtimeforstate =0011wewould have: H 0011 =1+[1 = 16 H 0011 +1 = 16 H 1011 +1 = 16 H 1111 +1 = 16 H 1001 +1 = 16 H 1010 +1 = 16 H 1101 +1 = 16 H 1110 +1 = 16 H 1000 +1 = 16 H 1100 +1 = 16 H 0111 +1 = 16 H 0101 +1 = 16 H 1001 +1 = 16 H 0100 +1 = 16 H 0001 +1 = 16 H 0000 +1 = 16 H 0010 ] UsingtheglobalstatediagramandtheactivityprobabilityPforeachstatewecan writetheequationsofhaltingforeachoftherandominitialstateasfollows: H 0011 =1+1 = 16[ H 0011 + H 1011 + H 1111 + H 1001 + H 1010 + H 1101 + H 1110 + H 1000 + H 1100 + H 0111 + H 0101 + H 0110 + H 0100 + H 0001 + H 0000 + H 0010 ] H 1011 =1+1 = 16[2 H 0011 +2 H 1011 +2 H 0001 +2 H 1000 +2 H 1010 +2 H 1001 +2 H 0010 +2 H 0000 ] 27

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H 1111 =1+1 = 16[ H 0011 + H 1011 + H 1111 + H 1001 + H 1010 + H 1101 + H 1110 + H 1000 + H 1100 + H 0111 + H 0101 + H 0110 + H 0100 + H 0001 + H 0000 + H 0010 ] H 1001 =0 H 1010 =1+1 = 16[ H 0011 + H 1011 + H 1111 + H 1001 + H 1010 + H 1101 + H 1110 + H 1000 + H 1100 + H 0111 + H 0101 + H 0110 + H 0100 + H 0001 + H 0000 + H 0010 ] H 1101 =1+1 = 16[2 H 1101 +2 H 0101 +2 H 0001 +2 H 0100 +2 H 0000 +2 H 1001 +2 H 1000 +2 H 1100 ] H 1110 =1+1 = 16[2 H 1110 +2 H 0110 +2 H 0010 +2 H 0100 +2 H 0000 +2 H 1010 +2 H 1000 +2 H 1100 ] H 1000 =1+1 = 16[2 H 1000 +2 H 1100 +2 H 1010 +2 H 1001 +2 H 1110 +2 H 1101 +2 H 1111 +2 H 1011 ] H 1100 =1+1 = 16[ H 0011 + H 1011 + H 1111 + H 1001 + H 1010 + H 1101 + H 1110 + H 1000 + H 1100 + H 0111 + H 0101 + H 1001 + H 0100 + H 0001 + H 0000 + H 0010 ] H 0111 =1+1 = 16[2 H 0111 +2 H 0011 +2 H 0101 +2 H 0110 +2 H 0001 +2 H 0010 +2 H 0100 +2 H 0000 ] H 0101 =1+1 = 16[ H 0011 + H 1011 + H 1111 + H 1001 + H 1010 + H 1101 + H 1110 + H 1000 + H 1100 + H 0111 + H 0101 + H 1001 + H 0100 + H 0001 + H 0000 + H 0010 ] H 1001 =0 H 0100 =1+1 = 16[2 H 0100 +2 H 1100 +2 H 1110 +2 H 1101 +2 H 1111 +2 H 0110 +2 H 0101 +2 H 0111 ] H 0001 =1+1 = 16[2 H 0001 +2 H 1001 +2 H 1101 +2 H 1011 +2 H 1111 +2 H 0101 +2 H 0111 +2 H 0011 ] H 0000 =1+1 = 16[ H 0011 + H 1011 + H 1111 + H 1001 + H 1010 + H 1101 + H 1110 + 28

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H 1000 + H 1100 + H 0111 + H 0101 + H 1001 + H 0100 + H 0001 + H 0000 + H 0010 ] H 0010 =1+1 = 16[2 H 0010 +2 H 1010 +2 H 1110 +2 H 1011 +2 H 1111 +2 H 0110 +2 H 0111 +2 H 0011 ] Weconsiderthisasasystemofequationswith14unknowns 4 where H x interprets astheexpectedhaltingtimeforstate x .Solvingthissystemofequationswillresultin anumericalvaluefortheexpectedhaltingtimeforeachof16possiblestates. H 0011 =8 : 031249990 H 1011 =8 : 028124995 H 1111 =8 : 093749991 H 1001 =0 : 0 H 1010 =8 : 031249986 H 1101 =8 : 028124997 H 1110 =8 : 028124997 H 1000 =8 : 034374998 H 1100 =8 : 031249992 H 0111 =8 : 028124996 H 0101 =8 : 031249992 H 0110 =0 : 0 H 0100 =8 : 034374996 H 0001 =8 : 034374991 H 0000 =8 : 031249992 H 0010 =8 : 034374998. Theaverageexpectedhaltingtimeiseasilycalculatedtobe8.035714279 Clearlychangingtheprobabilityofactivityforeachcellwillchangethetransitionprobability.Thisbringsupafewmoreinterestingquestions: Howwillchangingcellactivityprobabilityaectthetransitionprobability? Willchangingthetransitionprobabilityaecttheexpectedhaltingtime? Howwillthechangeintransitionprobabilityaecttheexpectedhaltingtime? Toanswerthesequestionswewritethehaltingequationforeachoftherandominitial statesleavingthecellactivityprobabilityasavariablep.Tondtheminimumhalting timewedierentiatetheequationagainwithpstayingasavariable.Theideaistowrite genericequationsbasedontheformulabottomofpage22andequationsshownonpages 4 Thereareonly14unknownsbecause H 0110 and H 1001 arezero 29

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23-24,exceptthatnottheprobabilityofactivityforeachcellisleftasvariablep.To deliveraclearerpointlet'slookatthefollowingexample:Supportweareattherandom initialstate[0011]andwewanttocomputetheexpectedhalting.Thebaseformulais: H =1+ X P H Sointhiscase is0011.Tondthe p wewillrefertotheglobalstatediagramto seehow0011willchangetootherstatesandndthosetransitionprobabilitiesbasedon probabilityofactivity: 30

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Figure20: Transitionprobabilitiesbasedonprobabilityofactivityasavariablep Asexplainedpreviouslyonpages21-22,towritethetransitionprobabilitywecombinetheprobabilityofactivityofthecellsfortherandominitialstate.Ifthecellisactive itsprobabilityofactivityispandifitisinactivetheprobabilityofactivityis-p.The gureaboveshowsallthetransitionprobabilitiesfrominitialstate[0011]toallother possiblenextstatesincludingitself.Nowitshouldbeverysimpletondtheexpected haltingtimeforthegiveninitialstate[0011].Tondtheexpectedhaltingtimeinterms ofpwecanwrite: H 0011 = )]TJ/F23 11.9552 Tf 10.4 0 Td [(p 4 [ H 0011 +1]+ p )]TJ/F23 11.9552 Tf 10.399 0 Td [(p 3 [ H 1011 +1]+ p )]TJ/F23 11.9552 Tf 10.399 0 Td [(p 3 [ H 0111 +1]+ p )]TJ/F23 11.9552 Tf 10.4 0 Td [(p 3 [ H 0001 +1] + p )]TJ/F23 11.9552 Tf 11.177 0 Td [(p 3 [ H 0010 +1]+ p 2 )]TJ/F23 11.9552 Tf 11.178 0 Td [(p 2 [ H 1111 +1]+ p 2 )]TJ/F23 11.9552 Tf 11.177 0 Td [(p 2 [ H 1001 +1]+ p 2 )]TJ/F23 11.9552 Tf 11.177 0 Td [(p 2 [ H 1010 +1] 31

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+ p 2 )]TJ/F23 11.9552 Tf 11.177 0 Td [(p 2 [ H 0101 +1]+ p 2 )]TJ/F23 11.9552 Tf 11.177 0 Td [(p 2 [ H 0110 +1]+ p 2 )]TJ/F23 11.9552 Tf 11.177 0 Td [(p 2 [ H 0000 +1]+ p 3 )]TJ/F23 11.9552 Tf 11.177 0 Td [(p [ H 1101 +1] + p 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p [ H 1110 +1]+ p 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p [ H 1000 +1]+ p 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p [ H 0100 +1]+ p 4 [ H 1100 +1] 32

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. Letusdemonstratethispointwithanothergureandexample,wherenowtheinitial statehasastablecell: Figure21: Transitionprobabilitiesforaninitialstatewithstablecell similarlytondtheexpectedhaltingtimeintermsofpforinitialstate[1101]wehave: H 1101 =[ )]TJ/F23 11.9552 Tf 11.955 0 Td [(p 4 + p )]TJ/F23 11.9552 Tf 11.955 0 Td [(p 3 ][ H 1101 +1]+[ p )]TJ/F23 11.9552 Tf 11.955 0 Td [(p 3 + p 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p 2 ][ H 0101 +1] +[ p )]TJ/F23 11.9552 Tf 11.955 0 Td [(p 3 + p 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(p 2 ][ H 1001 +1]+[ p )]TJ/F23 11.9552 Tf 11.955 0 Td [(p 3 + p 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p 2 ]+[ H 1100 +1] +[ p 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p 2 + p 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p ][ H 0001 +1]+[ p 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p 2 + p 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p ][ H 0100 +1] +[ p 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(p 2 + p 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p ][ H 1000 +1]+[ p 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p + p 4 ][ H 0000 +1] Nowwecanwritesimilarequationscorrespondingtoeachofthe16initialstatesand 33

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ndtheexpectedhaltingtimeforeachinitialstateintermsofpasavariable.Wecan thensettheseequationsasasystemwith16unknowns.Oncetheequationissolved wewillbeleftwithonenalequationintermsofpasavariable.Clearlywecannd theminimumvalueforthatequationbydierentiatingwithrespecttopandndingthe criticalpoints.NextwedeviseaMapleprogramtocalculatethehaltingtimewith P beingasimplevariableandallotherfactorsxed.Asimpleloopissetuptochangethe P startingfrom0.999goingdownto0.001steppingevery0.001.Thenthisexperimental datasetisgraphedagainstthecalculatedhaltingtime.Theresultisquiteamazing.The followinggureshowsthatastheprobabilityofactivityapproaches0.4thehaltingtime becomeslower. Figure22: Haltingtimevs.probabilityofactivity. Toseetheresultsmoreclearlyletuspresentthenextgure: 34

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Figure23: Closeupofhaltingtimevs.probabilityofactivity. Bysolvingtheequationswendthatat P =0 : 406thehaltingtimeisthelowest haltingtimeforthe2PartitioNsquareasynchronousnetworkbeingpreciselyat7 : 5838286. Interestinglyenoughthishaltingtimeiscertainlylowerthantheonewecalculatedusing the16equationsexploredearlier.Inadditionthislowerhaltingtimewasachievedusing alowerprobabilityofactivitythan P =1 = 2.Itisalsoimportanttonotethatwhenthe probabilityofactivityforthecellsiscloseto1,meaningeachcellisactiveatalltimes, thehaltingtimeapproachesinnity.Inotherword,itwilltakethisnetworkforeverto halt. 35

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6Philosophyandconclusion Theresultsobtainedinprevioussectionarenotwhatwewouldexpect.Imaginewith thisnetworkwasasimplecomputerwithfourprocessors,tryingtosolveaproblem.In realitybyincreasingtheactivityofeachprocessoronewouldexpecttosolvetheproblem faster,butinthiscasewerealizedthatslowingdowneachprocessorjustenough,would makethesimplecomputersolvetheproblemfaster.Letusconsiderascenariowhere wehavenitelymanyofthese2PartitioNsquareasynchronousnetworksinalargegrid. Each2PartitioNsquarenetworkhasfourlinkstofourother2PartitioNnetwork.Allthe squaresareatrestinarandominitialstate.Randomlyweactivateoneofthe2PartitioN squarenetworks.Assoonasthisonenetworkreachesthehaltingstate,itactivatesitsfour linkstootherfourneighboringnetworks.Theneachofthosefourneighboringnetworks becomeactiveandbegintoworktheirwaytowardshalting.Thisprocesscontinuesfor aniteperiodoftime,untilallthe2PartitioNsquaresarehaltedintheentiregrid. Thisprocessverymuchsimulatesthehealingprocessofanorganortissueinaliving organism.Supposeeach2PartitioNsquarenetworkinthisgridissetupwithaprobability ofactivityas P =1 = 2.Thenwecaneasilypredicttheaveragetimeforgivennumber ofcellstoreachtheirgoal,inthiscasehealingoftheentiretissue.Obviouslyifthereis aspecialsubstancethatcanlowertheprobabilityofactivityineachcell,thusslowing downthe2PartitioNsquarenetworkstothedesiredlevel{rememberthe P =0 : 406{then wecancertainlyachieveasignicantlyfasterhealingprocesswhenworkingwithlarge tissuesconsistingofmillionsofcells.Sofarquiteafewmathematicianshaveattempted toprovidetheclosestandmostidealmathematicalmodeltocellinteractionandgrowth asseeninnature.Theapproachtothismathematicalmodelmayseemsimple;however itprovidesabridgebetweentraditionalcomputationmodelsandnewapproachesto understandingcellularnetworksinnature.Wehaveallobservedtheusualcomputation modelsinanyregularcomputingmachine,anythingfromasimpleTuringmachineto afastsupercomputer.Theideahasalwaysbeentousehighlyregularandorganized modelswithstrictsetofrulesgoverningthemodelanditselements.Today'sscience 36

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howeverdemandsonetobreakfreeofregularityandorder,tolookatamodelseemingly overwhelmedinchaos,butwithjustenoughordertorepresentasophisticatedmodelof cellularactivityinnature. 37

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References Brzozowski,J.A.andR.Negulescu, AutomataofAsynchronousBehaviors,TheoreticalComputerScience,Vol.231,Issue1,pp.113-128,17January2000. Dijkstra, Self-stabilizingsystemsinspiteofdistributedcontrol,CACMpp.643-645, n.11,v.17,1974. H.BersiniandV.Detours, 1994.Asynchronyinducesstabilityincellular automatabasedmodels,ProceedingsoftheIVthConferenceonArticialLife,pages 382-387,Cambridge,MA,July1994,vol204,no.1-2,pp.70-82. Nehaniv,C.L. 2004AsynchronousAutomataNetworksCanEmulateAnySynchronousAutomataNetwork,InternationalJournalofAlgebraandComputation,146:719-739. StarkW.R. AsynchronousNetworksofAutomata-AStudyofEmergentPhonomenafromMathematicalAnalysistoBiologicallyInspiredApplications,Departmentof MathematicsandStatistics-UniversityofSouthFlorida,TampaFlorida Strogatz,Steven, 2003,SYNC,TheemergingScienceofSpontaneousOrder,1st Edition,NewYork vonNeumann,John, 1966,TheTheoryofSelf-reproducingAutomata,A.Burks, ed.,Univ.ofIllinoisPress,Urbana,IL. 38

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Ardestani, Arash Khani.
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Asynchronous cellular automata special networks local slowdown produces global speedup
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ABSTRACT: Information processing in living tissues is dramatically different from what we see in common man-made computer. The data and processing is distributed into the activity of cells which communicate only with neighboring cells. There is no clock for the global synchronization of cellular activities. There is not even one bit of central memory for globally shared data. The communication network between cells is highly irregular and may change without changing the outcome of the computation. A simple network of automata is introduced and analyzed to represent a mathematical model of special group of cells in an imaginary tissue sample. The interaction between the cells, their communication method, and their level of intelligence is studied. Three different structures of this model are demonstrated. Later on a simplification in the cells' program and elimination of a beat keeping clock will lead to a finite state automata network that is surprisingly more powerful in achieving the overall network's goal than its previous generation which had the advantage of more complex programs and a beat keeping clock.
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