Reaction-Diffusion Automata: Phenomenology, Localisations, Computation

327 Pages·2013·77.98 MB·English

by  Andrew Adamatzky (auth.)

Emergence, Complexity and Computation 1
SeriesEditors
Prof.IvanZelinka
TechnicalUniversityofOstrava
CzechRepublic
Prof.AndrewAdamatzky
UnconventionalComputingCentre
UniversityoftheWestofEngland
Bristol
UnitedKingdom
Prof.GuanrongChen
CityUniversityofHongKong
Forfurthervolumes:
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Andrew Adamatzky
Reaction-Diffusion
Automata: Phenomenology,
Localisations, Computation
ABC
Author
Prof.AndrewAdamatzky
UnconventionalComputingCentre
andDepartmentofComputerScience
UniversityoftheWestofEngland
Bristol
UK
ISSN2194-7287 e-ISSN2194-7295
ISBN978-3-642-31077-5 e-ISBN978-3-642-31078-2
DOI10.1007/978-3-642-31078-2
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012940855
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Preface
Cellularautomataareregularuniformnetworksoflocally-connectedfinite-statema-
chines,orcells.Cellstakediscretestatesandupdatetheirstatessimultaneouslyin
discrete time. Each cell chooses its next state depending on states of its closest
neighbours.The cell-state transition rules are very simple and intuitive yet allows
forcodinganon-trivialspace-timedynamics.Thusthecellularautomataisanideal
toolforafast-prototypingofnon-linearmediamodels,massive-parallelcomputers
andmathematicalmachines.Usingcellular-automatonmodelsofreaction-diffusion
andexcitablesystemsweanalysephenomenologyspatialdynamicsandshowhow
to implement computation in these automaton models of a non-linear media. We
complementcellular-automatonmodelswithautomataonplanarproximitygraphs.
Thebookconsistsofthreeparts.Inthefirstpartweintroducereaction-diffusion
and excitable cellular automata and automata on proximity graphs, study phe-
nomenology of propagating patterns, and represent a ’zoo’ of travelling and
stationarylocalizations.Reaction-diffusionisrepresentedin termsofinter-species
interactions in automaton models of populations in the second part. There we
discuss dynamics and complexity of inter-species interactions and analyse mutu-
alistic relationships. Computation in reaction-diffusion and excitable automata is
overviewedinthethirdpart.Therewedescribeautomatonnetworkswhichexecute
a space tessellation, we demonstrate how adders and multipliers are implemented
by colliding gliders in excitable medium, we show how operate binary strings
in reaction-diffusion automata and overview minimalistic models of memristive
networks.
The book is self-consistent and does not require any special knowledge to ap-
prehend. All models are intuitive and can be implemented with minimal knowl-
edge of programming.Abundantillustrations help to appreciate expressive power
ofdynamicalsystemson cellularautomataandplanarautomatonnetworks.Ideas,
implementationsandanalysisofferedwillattractreadersfromallwalksoflife:ev-
eryoneintriguedbysophisticatedbehaviourofcellularautomata,non-linearmedia
andmathematicalmachines.
AndrewAdamatzky
Bristol
Acknowledgements
Iamthankfulandgratefulto
• GenaroMartinezforcontributingtoChapter2bydiscussingphenomenological
classification of binary-state reaction-diffusion automata and co-authoring pa-
per[16].
• EmmanuelSapinforevolvingcell-statetransitionsmatricesofreaction-diffusion
automatadiscussedinChapter8andselectingtherulesrichintravellinglocali-
sations[20].
• MartinGrubeforaddingbiologicalflavourtoChapters9and10.
• LiangZhangforadvancingmyideasoncollision-basedbasedbinaryarithmetics
andoriginalimplementationof 2+-mediumonebithalf-adder;mostdesignsof
thechapterarebasedonpapersco-authoredwithLiang[281,282].
• AndyWuenschefortellingmeabouthisbeehiverule,whichpushedmetodis-
coverthespiralruleautomata[17],andforcontributingtoChapter13
• Leon Chua for introducing me to memristors which inspired some automaton
constructsinChapters14and15.
• ThomasDitzinger,EngineeringEditorial,Springer-Verlag,forbeingsosupport-
iveandhelpful.
Contents
1 Introduction................................................... 1
1.1 WeirdModsofExcitableAutomata ........................... 3
1.2 ExcitationonProximityGraphs .............................. 6
1.3 AutomatedSearchforLocalisations ........................... 8
1.4 Reaction-Diffusion,AutomataandPopulations.................. 8
1.5 MinimalModelsofPopulationDynamics ...................... 9
1.6 TowardsComputinginReaction-DiffusionAutomata ............ 11
1.7 Collision-BasedComputinginExcitableAutomata .............. 13
1.8 SpiralRuleAutomata ....................................... 14
1.9 MemristorsinCellularAutomata ............................. 16
1.10 OnColors................................................. 18
PartI PhenomenologyandLocalisations
2 Reaction-DiffusionBinary-StateAutomata........................ 21
2.1 PrecipitatingAutomata...................................... 22
2.2 Diffusion-AssociationAutomaton............................. 25
2.3 FunctionalClassification .................................... 29
2.4 ExcitableAutomatawithoutRefractoryState ................... 29
2.5 Summary ................................................. 34
3 RetainedExcitation ............................................ 37
3.1 RectangularlyGrowingDomains ............................. 38
3.2 Diamond-ShapedGrowingDomains .......................... 42
3.3 OctagonallyGrowingDomains ............................... 48
3.4 AlmostDisc-ShapedGrowingDomains........................ 52
3.5 AmoeboidGrowthofMixed-StatePatterns ..................... 53
3.6 NotGrowingDomainsofExcitation........................... 56
3.7 DomainswithSmallNumberofStillLocalizations .............. 61
3.8 MobileLocalizations ....................................... 65
3.9 Summary ................................................. 65
XII Contents
4 MutualisticExcitation .......................................... 67
4.1 PhenomenologyofMutualisticExcitation ...................... 68
4.2 MobileLocalisations........................................ 70
4.3 StationaryLocalisations ..................................... 80
4.4 HugeLocalisations ......................................... 88
4.5 CharacterisingLocalisations ................................. 92
4.6 ExcitationRulesRichwithLocalisations....................... 93
4.7 Summary ................................................. 94
5 DynamicalExcitationIntervals:DiversityandLocalisations ........ 97
5.1 MorphologicalDiversity..................................... 99
5.2 GenerativeDiversityandLocalisations.........................107
5.3 Summary .................................................113
6 ExcitableDelaunayTriangulations...............................115
6.1 StructuralPropertiesofDelaunayAutomata ....................117
6.2 AbsoluteExcitability .......................................121
6.3 RelativeExcitation .........................................123
6.4 Summary .................................................133
7 Excitableβ-Skeletons...........................................135
7.1 AbsolutelyExcitableSkeletons...............................137
7.2 RelativelyExcitableSkeletons................................141
7.3 StabilityofLocalisedOscillators..............................151
7.4 Summary .................................................151
8 EvolvingLocalizationsinReaction-DiffusionAutomata ............155
8.1 BreedingGlider-SupportingRules ............................156
8.2 LikehoodofGliders ........................................156
8.3 Quasi-chemicalReaction ....................................158
8.4 ReductionsofTransitionsFunctions...........................161
8.5 Summary .................................................162
PartII PopulationDynamics
9 PopulationDynamicsinAutomata ...............................165
9.1 Mutualism ................................................166
9.2 CommensalismandAmensalism..............................168
9.3 Parasitism.................................................169
9.4 Competition...............................................177
9.5 Summary .................................................180
10 AutomatonMechanicsofMutualism .............................183
10.1 Phenomenology............................................184
10.2 LocalisationsinMutualisticSystems ..........................185
10.3 Summary .................................................193
Contents XIII
PartIII ComputationwithExcitation
11 VoronoiAutomata..............................................199
11.1 VoronoiAutomata..........................................199
11.2 ConstructingVoronoiDiagramonVoronoiAutomata ............202
11.3 Arbitrary-ShapedPlanarObjectsandContours..................202
11.4 Summary .................................................207
12 AddersandMultipliersinSub-excitableAutomata.................209
12.1 Adders ...................................................210
12.2 Multipliers ................................................216
12.3 Summary .................................................228
13 ComputinginHexagonalReaction-DiffusionAutomaton ...........229
13.1 InputInterface .............................................233
13.2 MemoryDevice............................................235
13.3 RoutingandTuningSignals..................................237
13.4 BinaryOperations..........................................240
13.5 ImplementationoftheFiniteStateMachine.....................242
13.6 TransformationofTwo-andFour-BitStrings ...................244
13.7 SixBitCoding.............................................250
13.8 ScyllaandCharybdis:OutcomesofPassingbetweenTwoEaters...252
13.9 Summary .................................................255
14 Semi-memristiveAutomata:RetainedRefractoriness ..............263
14.1 Methods:ExperimentsandClassificiation ......................264
14.2 Classes ...................................................264
14.3 Hierarchies................................................270
14.4 TravellingLocalisations .....................................277
14.5 Summary .................................................283
15 StructuralDynamics:MemristiveExcitableAutomata .............287
15.1 Phenomenology............................................289
15.2 OscillatingLocalisations ....................................296
15.3 DynamicsofExcitationonInterfaces..........................300
15.4 BuildingConductivePathways ...............................302
15.5 Summary .................................................308
15.6 Appendix .................................................309
Epilogue ..........................................................311
References.........................................................313
Index .............................................................327
Chapter 1
Introduction
Cellularautomataareregularuniformnetworksoflocally-connectedfinite-statema-
chines, called cells. A cell takes a finite number of states. Cells are locally con-
nected:everycellupdatesitsstatedependingonstatesofitsgeographicallyclosest
neighbours.All cells updatetheir states simultaneouslyin discrete time steps. All
cellsemployethesameruletocalculatetheirstates.Cellularautomataarediscrete
systemswithnon-trivialbehaviour.Theyaremathematicalmodelsofcomputation
and computer models of natural systems. The cellular automata forms theoretical
backgroundand,atthesame timesimulationtoolsandimplementationsubstrates,
ofmathematicalmachineswithunboundedmemory,discretetheoreticalstructures,
digital physics and modelling of spatially extended non-linear systems; massive-
parallel computing,languageacceptance,and computability;reversibilityof com-
putation, graph-theoretic analysis and logic; chaos and undecidability; evolution,
learningandcryptography.Itisalmostimpossibletofindafieldofnaturalandtech-
nical sciences, where cellular automata are not used. For those not familiar with
cellular automata we recommend to have a look in few classical titles. You can
startwithaToffoli-Margolus’sbestseller[242]andthenspoilyourselfwithlavishly
illustratedatlasbyWuensche[266]andthought-provokingcellularautomatontrea-
tise by Wolfram [262]. Plenty of interesting state transition rules and useful hints
and tips on can be found in Ilachinski’s compendium of cellular-automaton uni-
verse [144]. Conway’s Game of Life is the most popular cellular automaton, we
recommendthecollectionofchaptersasatreatise[25]oftheGame-ofLifeinves-
tigations, approachesand findings. Comprehensivespecialised texts on modelling
space-timedynamicsofnaturalprocessesincellularautomataareauthoredbyBoc-
cara[56],ChopardandDroz[77],Weimar[258]andDeutschandDormann[95].
Since their inception in [122], cellular automaton models of excitation be-
came a usual tool for studying complex phenomena of excitation wave dynamics
and chemicalreaction-diffusionactivitiesin physical,chemicaland biologicalsys-
tems [77, 144]. They are now essential instruments in computational analysis of
non-linearsystems,andexhaustivesearchfornon-trivialfunctionsincell-statetran-
sition rule spaces [9], [16]. Cellular-automaton models of reaction-diffusion and
excitable systems are of particular importance because by using them we can —
A.Adamatzky:Reaction-DiffusionAutomata,ECC1,pp.1–18.
springerlink.com (cid:2)c Springer-VerlagBerlinHeidelberg2013