Cellular Automata: Analysis and Applications

467 Pages·2017·7.344 MB·English

by  Karl-Peter Hadeler, Johannes Müller (auth.)

Springer Monographs in Mathematics
Karl-Peter Hadeler
Johannes Müller
Cellular
Automata:
Analysis and
Applications
Springer Monographs in Mathematics
Editors-in-Chief
IsabelleGallagher
MinhyongKim
Moreinformationaboutthisseriesathttp://www.springer.com/series/3733
Karl-Peter Hadeler • Johannes MuRller
Cellular Automata: Analysis
and Applications
123
Karl-PeterHadeler JohannesMuRller
FBBiologie CentreforMathematicalSciences
UniversitaRtTuRbingen TechnicalUniversityMunich
TuRbingen,Germany Garching,Germany
ISSN1439-7382 ISSN2196-9922 (electronic)
SpringerMonographsinMathematics
ISBN978-3-319-53042-0 ISBN978-3-319-53043-7 (eBook)
DOI10.1007/978-3-319-53043-7
LibraryofCongressControlNumber:2017937054
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Contents
1 Introduction................................................................. 1
1.1 Discreteness .......................................................... 1
1.2 TheGameofLife .................................................... 3
1.3 ContactAutomata.................................................... 4
1.4 SomeWolframAutomata............................................ 6
1.5 Greenberg-HastingsAutomata ...................................... 10
1.6 Langton’sAntandLifeWithoutDeath ............................. 11
1.7 ANiceLittleAutomaton............................................. 12
1.8 HistoryandApplications............................................. 13
1.9 OutlineofThisWork................................................. 13
2 CellularAutomata:BasicDefinitions .................................... 19
2.1 TheGrid .............................................................. 19
2.1.1 AbelianorRegularGrids................................... 20
2.1.2 Non-AbelianGrids ......................................... 22
2.2 TheNeighborhood ................................................... 26
2.3 ElementaryStateandtheGlobalState.............................. 28
2.4 TheLocalandtheGlobalFunction.................................. 30
2.5 Excursion:TheGrowthFunctionofaCayleyGraph............... 33
3 CantorTopologyofCellularAutomata.................................. 37
3.1 Prelude:CantorSetsandCantorSpaces ............................ 38
3.1.1 TheClassicalMid-ThirdCantorSet....................... 38
3.1.2 CantorSpaces............................................... 42
3.2 CantorMetricforCellularAutomata................................ 45
3.3 TheCurtis-Hedlund-LyndonTheorem.............................. 48
3.4 SpatialStructureandSimplifications................................ 52
3.4.1 Examples:StructuresThatAreNotCellular
Automata.................................................... 57
3.4.2 SimplificationoftheStateSpace .......................... 60
v
vi Contents
3.4.3 SimplificationoftheNeighborhood....................... 61
3.4.4 SimplificationoftheGrid.................................. 62
3.5 CellularAutomataandContinuousMapsonCantorSpaces....... 65
3.5.1 BijectiveMaps.............................................. 66
3.5.2 GeneralMaps:TheUniversalCellularAutomaton....... 67
4 BesicovitchandWeylTopologies.......................................... 75
4.1 DefinitionoftheBesicovitchandWeylSpace...................... 75
4.2 TopologicalProperties ............................................... 81
4.2.1 BesicovitchSpaces ......................................... 82
4.2.2 WeylSpaces................................................. 90
4.3 CellularAutomataonBesicovitchandWeylSpaces............... 98
4.4 ACHLTheoremforBesicovitchandWeylSpaces ................ 103
5 Attractors.................................................................... 111
5.1 DynamicalSystems,!-LimitSetsandAttractors.................. 112
5.1.1 DynamicalSystems......................................... 112
5.1.2 !-LimitSetsandAttractors................................ 114
5.2 StructureofAttractors:FiniteGrids................................. 119
5.3 IntersectionofAttractorsandQuasi-Attractors..................... 119
5.4 ConleysDecompositionTheorem,Attractors,andChains......... 125
5.5 BernoulliMeasureonCellularAutomata........................... 132
5.6 StructureofAttractors—InfiniteGrids:HurleyClassification..... 140
6 ChaosandLyapunovStability ............................................ 155
6.1 TopologicalChaos.................................................... 155
6.2 PermutingCellularAutomata........................................ 159
6.2.1 SurjectiveCellularAutomata .............................. 160
6.2.2 TopologicalTransitivity.................................... 165
6.2.3 DensenessofPeriodicPoints .............................. 166
6.3 LyapunovStabilityandGilmanClassification...................... 169
6.3.1 ClassGilman1.............................................. 171
6.3.2 ClassGilman2.............................................. 173
6.3.3 ClassGilman3.............................................. 174
6.3.4 ClassGilman4.............................................. 175
7 LanguageClassificationofKu˚rka ........................................ 179
7.1 Grammar.............................................................. 179
7.2 FiniteAutomata ...................................................... 181
7.3 FiniteAutomataandRegularLanguages ........................... 184
7.4 CellularAutomataandLanguage:Ku˚rkaClassification ........... 186
7.4.1 ClassKu˚rka1............................................... 190
7.4.2 ClassKu˚rka2............................................... 191
7.4.3 Ku˚rka3...................................................... 194
8 TuringMachines,Tiles,andComputability............................. 197
8.1 TuringMachines ..................................................... 197
8.2 UniversalTuringMachine ........................................... 201
Contents vii
8.3 ComputationalUniversalityofCellularAutomata ................. 205
8.4 UndecidableProblems............................................... 207
8.4.1 TheParadoxofEpimenides................................ 207
8.4.2 Russel’sParadox............................................ 208
8.4.3 Richard’sParadox .......................................... 209
8.4.4 TheWordProblem ......................................... 210
8.4.5 TheHaltingProblem ....................................... 211
8.4.6 TheImmortalityProblem .................................. 213
8.4.7 Non-computabilityof!-LimitSetsforCellular
Automata.................................................... 213
8.5 Tiles................................................................... 214
8.5.1 DefinitionsandExamples.................................. 214
8.5.2 TessellationsofFreeGroups............................... 217
8.5.3 AperiodicTessellationsonZ2.............................. 220
8.5.4 UndecidabilityoftheDominoProbleminZ2............. 232
8.5.5 UndecidabilityoftheFiniteDominoProbleminZ2...... 239
8.5.6 GrouporGraph............................................. 246
8.5.7 DominoProblemandMonadicSecondOrderLogic..... 249
9 SurjectivityandInjectivityofGlobalMaps ............................. 253
9.1 TheGardenofEden.................................................. 254
9.2 AlgorithmsforOne-DimensionalCellularAutomata.............. 262
9.2.1 StationaryPoints............................................ 262
9.2.2 Surjectivity.................................................. 267
9.2.3 InjectivityandBijectivity .................................. 275
9.3 UndecidabilityHigherDimensionalCellularAutomata ........... 279
9.3.1 StationaryPoints............................................ 280
9.3.2 Surjectivity.................................................. 280
10 LinearCellularAutomata................................................. 287
10.1 RepresentationofLinearCellularAutomata........................ 287
10.2 Surjectivity,InjectivityandBijectivity.............................. 292
10.3 FractalSetsandLinearCellularAutomata.......................... 296
10.3.1 IntroductoryExampleandtheFermatProperty........... 297
10.3.2 LimitSetsofLinearCellularAutomata................... 300
10.3.3 IteratedFunctionSystems.................................. 307
10.3.4 MatrixSubstitutionSystems............................... 314
10.3.5 CellularAutomataandMatrixSubstitutionSystems..... 324
11 ParticleMotion ............................................................. 335
11.1 ParticleMotion:FormalApproach.................................. 335
11.1.1 ModellingDiffusionbyContinuousModels.............. 336
11.1.2 NaiveCellularAutomataModelsforDiffusion........... 338
11.2 FromPDEtoCellularAutomata:UltradiscreteLimit.............. 340
11.2.1 HeatEquation............................................... 342
11.2.2 TheBurgersEquation ...................................... 343
11.2.3 UltradiscreteLimitandBurgersEquation................. 350
viii Contents
11.3 MicroscopicModelsforDiffusion .................................. 354
11.3.1 StraightMovement ......................................... 355
11.3.2 LatticeGasCellularAutomata............................. 360
12 PatternFormation.......................................................... 377
12.1 FractalMolluscPatterns ............................................. 377
12.2 TuringPattern ........................................................ 378
12.2.1 Turing-PatterninPartialDifferentialEquations .......... 378
12.2.2 Excursion:HopfieldNets................................... 380
12.2.3 Bar-Yam-ModelforTuringPattern........................ 384
12.3 Greenberg-HastingsModelforExcitableMedia ................... 386
12.3.1 Definitions .................................................. 387
12.3.2 TheWindingNumber ...................................... 391
12.3.3 ThePotential................................................ 396
12.3.4 SurvivalofConfigurations................................. 400
13 ApplicationsinVariousAreas............................................. 405
13.1 SandpileAutomataandSelf-OrganizedCriticality................. 405
13.2 Epidemiology......................................................... 411
13.2.1 MeanFieldApproximation ................................ 411
13.2.2 SIRSModelandMeanFieldApproximation............. 413
13.2.3 PolynomialGrowth:ClusteringofContactNetworks.... 416
13.3 Evolution ............................................................. 418
13.3.1 Evolution.................................................... 419
13.3.2 SpatialModel............................................... 420
13.3.3 HeuristicAnalysis.......................................... 423
A BasicMathematicalTools.................................................. 427
A.1 BasicDefinitionsfromTopology.................................... 427
A.2 BasicAlgebraicTheory.............................................. 433
A.2.1 GroupTheory............................................... 433
A.2.2 RingTheory................................................. 438
A.2.3 Fields........................................................ 441
A.3 BasicMeasureTheory ............................................... 442
References......................................................................... 455
Index............................................................................... 463
List of Symbols
Wegivealistofsomenotationswithareferencetothefirstpagewherethenotation
appears.
Symbol Page Meaning
Œa(cid:2),Œab(cid:2) 162 Concatenationofsignsor(finite)sequences
Œa(cid:2),Œ0(cid:2) 151 StateinEZthatisidenticallya(or0)
kzk1 25 `1-normofavectorz
kzk 27 `1-normofavectorz
1
ŒGWH(cid:2) 29 IndexofasubgroupH ofG
jXj 33 SizeofsomesetX
0 26 Neutralelementofagroup,originofthegrid
(Abeliancase)
A 116 Conleyattractor
A 120 Setofallattractors
A.f/ 55 Setofbijectivefunctionscommutingwithf
B .x/ 46 Openballaroundxwithradiusrinametricspace
r
B.:/ 117,131 Basinofattraction(forattractorsresp.chain
components)
B 134 Borelalgebra
C 52 Cantorspaceormetric,compact,andtotally
disconnectedspace
C1=3 39 Themid-thirdCantorset
CR.f/ 125 Recurrentpoints
d.u;v/ 45 CantordistancebetweentwostatesinE(cid:3)
d.n1;n2/ 387 UnsigneddistanceonZm
d .g;h/ 25 DistancebetweentwoelementsonaCayley
c
graph
ix