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Game of Life Cellular Automata Andrew Adamatzky Editor Game of Life Cellular Automata Editor Prof.AndrewAdamatzky UniversityoftheWestofEngland BristolBS161QY UnitedKingdom [email protected] ISBN978-1-84996-216-2 e-ISBN978-1-84996-217-9 DOI10.1007/978-1-84996-217-9 SpringerLondonDordrechtHeidelbergNewYork BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressControlNumber:2010928553 ©Springer-VerlagLondonLimited2010 Apartfromanyfairdealingforthepurposesofresearchorprivatestudy,orcriticismorreview,asper- mittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, storedortransmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublish- ers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicensesissuedbythe CopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbesentto thepublishers. Theuseofregisterednames,trademarks,etc.,inthispublicationdoesnotimply,evenintheabsenceofa specificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandthereforefree forgeneraluse. Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinformation containedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsoromissions thatmaybemade. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface In 1970 Martin Gardner spelled out the rules of a new solitaire game forged by John Horton Conway.1 An unparalleled combination of functional simplicity with behavioural complexity made Conway’s Game of Life the most popular cellular automaton of all time. We commemorate the Game of Life’s 40th birthday with a unique collection of works authored by renowned mathematicians, computer sci- entists, physicists and engineers. The superstars of science, academy and industry present their visions of the Game of Life cellular automaton, its extensions and modifications,andspatially-extendedsystemsinspiredbytheGame. The book covers hot topics in theory of computation, pattern formation, opti- mization,evolution,non-linearsciencesandmathematics.Academics,researchers, hobbyistsandstudentsinterestedintheGameofLifetheoryandapplicationswill findthismonographavaluableguidetothefieldofcellularautomataandexcellent supplementaryreading. Bristol,UK AndrewAdamatzky 1“...eachcellofthecheckerboard(assumedtobeaninfiniteplane)haseightneighboringcells, fouradjacentorthogonally,fouradjacentdiagonally.Therulesare: 1. Survivals.Everycounterwithtwoorthreeneighboringcounterssurvivesforthenextgenera- tion. 2. Deaths.Eachcounterwithfourormoreneighborsdies(isremoved)fromoverpopulation.Every counterwithoneneighborornonediesfromisolation. 3. Births.Eachemptycelladjacenttoexactlythreeneighbors—nomore,nofewer—isabirth cell.Acounterisplacedonitatthenextmove. ...allbirthsanddeathsoccursimultaneously.” MartinGardner,ThefantasticcombinationsofJohnConway’snewsolitairegame“life”.Scientific American223(October1970):120–123. v Contents 1 IntroductiontoCellularAutomataandConway’sGameofLife . . . 1 CarterBays 1.1 ABriefBackground . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 TheOriginalGliderGun . . . . . . . . . . . . . . . . . . . . . . 4 1.3 OtherGoLRulesintheSquareGrid . . . . . . . . . . . . . . . . 5 1.4 WhyTreatAllNeighborstheSame? . . . . . . . . . . . . . . . . 5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 PartI Historical 2 Conway’sGameofLife:EarlyPersonalRecollections . . . . . . . . 11 RobertWainwright 3 Conway’sLife . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 HaroldV.McIntosh 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 TheRulesoftheGame . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 StillLifes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 PeriodTwo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.5 Gliders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.6 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.7 GliderGuns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.8 PufferTrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.9 LifeonaTorus . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.10 CyclesofFinitePeriodicity . . . . . . . . . . . . . . . . . . . . 30 4 Life’sStillLifes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 HaroldV.McIntosh 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 CellularAutomata . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 StillLife . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 vii viii Contents 4.4 DeBruijnDiagram . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.5 FirstStage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.6 SecondStage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5 AZooofLifeForms . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 HaroldV.McIntosh 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2 (0,0,1)—StillLifes . . . . . . . . . . . . . . . . . . . . . . . 52 5.2.1 FirstLeveldeBruijnMatrix . . . . . . . . . . . . . . . 52 5.2.2 PowersoftheStillLifedeBruijnMatrix . . . . . . . . . 53 5.2.3 SampleStillLifeStrips . . . . . . . . . . . . . . . . . . 57 5.3 (0,1,1)—LongitudinalCreepers . . . . . . . . . . . . . . . . . 58 5.3.1 FirstLeveldeBruijnMatrix . . . . . . . . . . . . . . . 58 5.3.2 PowersoftheLongitudinaldeBruijnMatrix . . . . . . . 59 5.3.3 SampleStripswithLongitudinalMovement . . . . . . . 61 5.4 (1,0,1)—TransversalCreepers. . . . . . . . . . . . . . . . . . 62 5.4.1 FirstLeveldeBruijnMatrix . . . . . . . . . . . . . . . 62 5.4.2 PowersoftheTransversaldeBruijnMatrix . . . . . . . 62 5.4.3 SampleStripswithTransversalMovement . . . . . . . . 65 5.5 (1,1,1)—DiagonalCreepers . . . . . . . . . . . . . . . . . . . 66 5.5.1 FirstLeveldeBruijnMatrix . . . . . . . . . . . . . . . 66 5.5.2 PowersoftheDiagonaldeBruijnMatrix . . . . . . . . . 66 5.5.3 SampleStripswithDiagonalMovement . . . . . . . . . 68 PartII ClassicalTopics 6 GrowthandDecayinLife-LikeCellularAutomata . . . . . . . . . . 71 DavidEppstein 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.2 Life-LikeRules. . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.3 NaturalEvolutionorIntelligentDesign?. . . . . . . . . . . . . . 73 6.4 Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.5 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.6 TheLife-LikeMenagerie . . . . . . . . . . . . . . . . . . . . . 83 6.7 BeyondGrowthandDecay. . . . . . . . . . . . . . . . . . . . . 93 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7 TheB36/S125“2x2”Life-LikeCellularAutomaton . . . . . . . . . . 99 NathanielJohnston 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.2 AshandCommonPatterns . . . . . . . . . . . . . . . . . . . . . 100 7.3 OscillatorsandSpaceships . . . . . . . . . . . . . . . . . . . . . 102 7.4 AsaBlockCellularAutomaton . . . . . . . . . . . . . . . . . . 104 7.5 BlockOscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Contents ix Appendix:PatternCollection . . . . . . . . . . . . . . . . . . . 111 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8 ObjectSynthesisinConway’sGameofLifeandOtherCellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 MarkD.Niemiec 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.2 SimpleObjectSynthesis . . . . . . . . . . . . . . . . . . . . . . 116 8.3 ArtImitatesLife . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8.4 LifeImitatesArt . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.4.1 WishingfortheImpossible . . . . . . . . . . . . . . . . 117 8.4.2 AlteringtheLawsofPhysics . . . . . . . . . . . . . . . 119 8.4.3 TheButterflyEffect . . . . . . . . . . . . . . . . . . . . 119 8.5 IncrementalSynthesis . . . . . . . . . . . . . . . . . . . . . . . 120 8.5.1 BetterLivingThroughChemistry . . . . . . . . . . . . . 120 8.5.2 TheJoyofCooking . . . . . . . . . . . . . . . . . . . . 121 8.5.3 Fireworks . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.5.4 OpenHeartSurgery . . . . . . . . . . . . . . . . . . . . 124 8.6 SynthesisofMovingObjects. . . . . . . . . . . . . . . . . . . . 126 8.6.1 SpaceshipsandFlotillas . . . . . . . . . . . . . . . . . . 126 8.6.2 PufferTrains. . . . . . . . . . . . . . . . . . . . . . . . 127 8.6.3 GliderGuns . . . . . . . . . . . . . . . . . . . . . . . . 128 8.6.4 BreedersandOtherLargePatterns . . . . . . . . . . . . 129 8.7 OtherRules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8.7.1 B34/S34Life . . . . . . . . . . . . . . . . . . . . . . . 130 8.7.2 B2/S2Life . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.8 RemainingIssues . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8.8.1 GeneralProblems . . . . . . . . . . . . . . . . . . . . . 133 8.8.2 ProblemswithStill-Lifes . . . . . . . . . . . . . . . . . 133 8.8.3 ProblemswithPseudo-objects . . . . . . . . . . . . . . 133 8.8.4 ProblemswithOscillators . . . . . . . . . . . . . . . . . 134 8.8.5 ProblemswithSpaceshipsandPufferTrains . . . . . . . 134 8.8.6 ProblemswithGliderGuns . . . . . . . . . . . . . . . . 134 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 9 GlidersandGliderGunsDiscoveryinCellularAutomata . . . . . . 135 EmmanuelSapin 9.1 FormalisationsofCellularAutomata . . . . . . . . . . . . . . . 136 9.1.1 SetofCellularAutomata . . . . . . . . . . . . . . . . . 136 9.1.2 EvolutionofCellularAutomata . . . . . . . . . . . . . . 136 9.1.3 Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.1.4 NumberofAutomata . . . . . . . . . . . . . . . . . . . 137 9.1.5 GameofLife . . . . . . . . . . . . . . . . . . . . . . . 140 9.1.6 QuiescentState . . . . . . . . . . . . . . . . . . . . . . 141 9.2 DefinitionsofPatterns . . . . . . . . . . . . . . . . . . . . . . . 141 9.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 141 x Contents 9.2.2 PeriodicPattern . . . . . . . . . . . . . . . . . . . . . . 141 9.2.3 Glider . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 9.2.4 GliderGun . . . . . . . . . . . . . . . . . . . . . . . . 143 9.2.5 SymmetricPattern. . . . . . . . . . . . . . . . . . . . . 144 9.3 AutomataandPatterns . . . . . . . . . . . . . . . . . . . . . . . 145 9.3.1 AutomataAcceptingPatterns . . . . . . . . . . . . . . . 145 9.3.2 AutomataofE AcceptingPatterns . . . . . . . . . . . . 146 9.3.3 AutomataofI AcceptingPatterns . . . . . . . . . . . . 146 9.4 SearchingforGliders. . . . . . . . . . . . . . . . . . . . . . . . 148 9.4.1 Rational . . . . . . . . . . . . . . . . . . . . . . . . . . 148 9.4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 148 9.4.3 Gliders . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.5 SearchingforGliderGuns . . . . . . . . . . . . . . . . . . . . . 152 9.5.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 9.6 GunsofaSpecificGlider . . . . . . . . . . . . . . . . . . . . . 157 9.6.1 DifferentGuns . . . . . . . . . . . . . . . . . . . . . . 159 9.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 9.7 SynthesisandPerspectives . . . . . . . . . . . . . . . . . . . . . 162 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10 ConstraintProgrammingtoSolveMaximalDensityStillLife . . . . 167 GeoffreyChu,KarenElizabethPetrie,andNeilYorke-Smith 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 10.2 MaximumDensityStillLife . . . . . . . . . . . . . . . . . . . . 168 10.3 ConstraintProgramming . . . . . . . . . . . . . . . . . . . . . . 169 10.4 ConstraintProgrammingModelsforStillLife . . . . . . . . . . 170 10.5 SolvingStillLife . . . . . . . . . . . . . . . . . . . . . . . . . . 171 10.5.1 DensityasBoardSizeIncreases . . . . . . . . . . . . . 172 10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 PartIII Asynchronous,ContinuousandMemory-EnrichedAutomata 11 LargerthanLife’sExtremes:RigorousResultsforSimplifiedRules andSpeculationonthePhaseBoundaries . . . . . . . . . . . . . . . 179 KellieMicheleEvans 11.1 IntroductionandHistory . . . . . . . . . . . . . . . . . . . . . . 179 11.2 LargerthanLife:DefinitionandNotation . . . . . . . . . . . . . 181 11.3 InitialStates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 11.4 LtLDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 11.4.1 LocalConfigurations . . . . . . . . . . . . . . . . . . . 186 11.4.2 ErgodicClassificationsfortheInfiniteSystem . . . . . . 187 11.5 LtLRuleswithSimplifyingFeatures . . . . . . . . . . . . . . . 188 11.5.1 SymmetricLtLRules . . . . . . . . . . . . . . . . . . . 188 11.5.2 MonotoneLtLRules . . . . . . . . . . . . . . . . . . . 196 Contents xi 11.5.3 BirthorDeathOnlyLtLRules . . . . . . . . . . . . . . 203 11.6 SnapshotsfromtheBoundaries . . . . . . . . . . . . . . . . . . 205 11.6.1 Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . 205 11.6.2 Self-organizedBatik . . . . . . . . . . . . . . . . . . . 206 11.6.3 GlobalTilingsandWaterbedDynamics . . . . . . . . . 207 11.6.4 SlowConvergence. . . . . . . . . . . . . . . . . . . . . 208 11.6.5 SlowGrowth . . . . . . . . . . . . . . . . . . . . . . . 209 11.6.6 Ladders . . . . . . . . . . . . . . . . . . . . . . . . . . 211 11.6.7 BugswithTrails . . . . . . . . . . . . . . . . . . . . . . 214 11.6.8 BugLogic,Bosco,andOpenQuestions . . . . . . . . . 215 11.6.9 LtL’sBugs:Threshold-RangeScaling . . . . . . . . . . 217 11.7 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 11.8 ConclusionandLong-TermGoals . . . . . . . . . . . . . . . . . 220 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 12 RealLife . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 MarcusPivato 12.1 WhatHappensinRealLife . . . . . . . . . . . . . . . . . . . . . 224 12.1.1 RealLifevs.LargerthanLife . . . . . . . . . . . . . . . 227 12.1.2 StillLifes . . . . . . . . . . . . . . . . . . . . . . . . . 228 12.1.3 RobustnessofStillLifesintheHausdorffMetric . . . . 230 12.2 TheAnatomyofBugs . . . . . . . . . . . . . . . . . . . . . . . 230 12.3 OpenQuestions . . . . . . . . . . . . . . . . . . . . . . . . . . 232 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 13 VariationsontheGameofLife . . . . . . . . . . . . . . . . . . . . . 235 FerdinandPeper,SusumuAdachi,andJiaLee 13.1 AsynchronouslyTimedGameofLife . . . . . . . . . . . . . . . 236 13.2 GameofLifewithContinuousStates . . . . . . . . . . . . . . . 241 13.3 GameofLifewithanEnlargedNeighborhood . . . . . . . . . . 247 13.4 ConclusionsandDiscussion . . . . . . . . . . . . . . . . . . . . 253 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 14 DoesLifeResistAsynchrony? . . . . . . . . . . . . . . . . . . . . . . 257 NazimFatès 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 14.2 ABriefHistoryoftheProblem . . . . . . . . . . . . . . . . . . 258 14.3 AsynchronousLife . . . . . . . . . . . . . . . . . . . . . . . . 259 14.4 AssessingLife’sRobustnesstoAsynchrony . . . . . . . . . . . 260 14.4.1 FirstExperiment . . . . . . . . . . . . . . . . . . . . . 260 14.4.2 AQuantificationoftheChanges . . . . . . . . . . . . . 261 14.4.3 ASecond-OrderPhaseTransition. . . . . . . . . . . . . 262 14.5 InitialConditionsandAsymptoticBehaviour . . . . . . . . . . . 264 14.5.1 ExperimentalApproach . . . . . . . . . . . . . . . . . . 264 14.5.2 Mean-FieldAnalysis . . . . . . . . . . . . . . . . . . . 265 14.5.3 AClose-uponSmallInitialDensities . . . . . . . . . . 266

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Game of Life Cellular Automata PDF

576 Pages·2010·24.788 MB·English
by  Carter Bays (auth.), Andrew Adamatzky (eds.)
#Computers #Algorithms and Data Structures
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