Table Of ContentEncyclopedia of
Complexity and Systems Science Series
Editor-in-Chief: Robert A. Meyers
Andrew Adamatzky Editor
Cellular Automata
A Volume in the Encyclopedia of
Complexity and Systems Science,
Second Edition
Encyclopedia of Complexity and
Systems Science Series
Editor-in-Chief
RobertA.Meyers
The Encyclopedia of Complexity and Systems Science Series of topical
volumesprovidesanauthoritativesourceforunderstandingandapplyingthe
concepts of complexity theory together with the tools and measures for
analyzing complex systems in all fields of science and engineering. Many
phenomenaatallscalesinscienceandengineeringhavethecharacteristicsof
complexsystems,andcanbefullyunderstoodonlythroughthetransdisciplin-
aryperspectives,theories,andtoolsofself-organization,synergetics,dynam-
ical systems, turbulence, catastrophes, instabilities, nonlinearity, stochastic
processes, chaos, neural networks, cellular automata, adaptive systems,
genetic algorithms, and so on. Examples of near-term problems and major
unknowns that can be approached through complexity and systems science
include:thestructure,history,andfutureoftheuniverse;thebiologicalbasisof
consciousness;theintegrationofgenomics,proteomics,andbioinformaticsas
systemsbiology;humanlongevitylimits;thelimitsofcomputing;sustainabil-
ityofhumansocietiesandlifeonearth;predictability,dynamics,andextentof
earthquakes,hurricanes,tsunamis,andothernaturaldisasters;thedynamicsof
turbulentflows;lasersorfluidsinphysics;microprocessordesign;macromo-
lecular assembly in chemistry and biophysics; brain functions in cognitive
neuroscience; climate change; ecosystem management; traffic management;
and business cycles. All these seemingly diverse kinds of phenomena and
structure formation have a number of important features and underlying
structures in common. These deep structural similarities can be exploited to
transferanalyticalmethodsandunderstandingfromonefieldtoanother.This
uniqueworkwillextendtheinfluenceofcomplexityandsystemsciencetoa
muchwideraudiencethanhasbeenpossibletodate.
More information about this series at https://link.springer.com/bookseries/
15581
Andrew Adamatzky
Editor
Cellular Automata
A Volume in the Encyclopedia of
Complexity and Systems Science,
Second Edition
With425Figuresand20Tables
Editor
AndrewAdamatzky
UnconventionalComputingCentre
UniversityoftheWestofEngland
Bristol,UK
ISBN978-1-4939-8699-6 ISBN978-1-4939-8700-9(eBook)
ISBN978-1-4939-8701-6(printandelectronicbundle)
https://doi.org/10.1007/978-1-4939-8700-9
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Series Preface
TheEncyclopediaofComplexityandSystemScienceSeriesisamultivolume
authoritative source for understanding and applying the basic tenets of com-
plexity and systems theory as well as the tools and measures for analyzing
complexsystemsinscience,engineering,andmanyareasofsocial,financial,
andbusinessinteractions.Itiswrittenforanaudienceofadvanceduniversity
undergraduateandgraduatestudents,professors,andprofessionalsinawide
range of fields who must manage complexity on scales ranging from the
atomicandmoleculartothesocietalandglobal.
Complexsystemsaresystemsthatcomprisemanyinteractingpartswiththe
ability to generate a new quality of collective behavior through self-
organization, e.g., the spontaneous formation of temporal, spatial, or func-
tionalstructures.Theyarethereforeadaptiveastheyevolveandmaycontain
self-drivingfeedbackloops.Thus,complexsystemsaremuchmorethanasum
of their parts. Complex systems are often characterized as having extreme
sensitivitytoinitialconditionsaswellasemergentbehaviorthatarenotreadily
predictableorevencompletelydeterministic.Theconclusionisthatareduc-
tionist(bottom-up)approachisoftenanincompletedescriptionofaphenom-
enon.Thisrecognitionthatthecollectivebehaviorofthewholesystemcannot
be simply inferred from the understanding of the behavior of the individual
componentshasledtomanynewconceptsandsophisticatedmathematicaland
modeling tools for application to many scientific, engineering, and societal
issues that can be adequately described only in terms of complexity and
complexsystems.
ExamplesofGrandScientificChallengeswhichcanbeapproachedthrough
complexity and systems science include: the structure, history, and future of
theuniverse;thebiologicalbasisofconsciousness;thetruecomplexityofthe
genetic makeup and molecular functioning of humans (genetics and epige-
netics)andotherlifeforms;humanlongevitylimits;unificationofthelawsof
physics;thedynamicsandextentofclimatechangeandtheeffectsofclimate
change;extendingtheboundariesofandunderstandingthetheoreticallimits
ofcomputing;sustainabilityoflifeontheearth;workingsoftheinteriorofthe
earth;predictability,dynamics,andextentofearthquakes,tsunamis,andother
natural disasters; dynamics of turbulent flows and the motion of granular
materials;thestructureofatomsasexpressedintheStandardModelandthe
formulation of the Standard Model and gravity into a Unified Theory; the
structureofwater;controlofglobalinfectiousdiseases;andalsoevolutionand
quantification of (ultimately) human cooperative behavior in politics,
v
vi SeriesPreface
economics, business systems, and social interactions. In fact, most of these
issues have identified nonlinearities and are beginning to be addressed with
nonlinear techniques, e.g., human longevity limits, the Standard Model, cli-
mate change, earthquake prediction, workings of the earth’s interior, natural
disasterprediction,etc.
The individual complex systems mathematical and modeling tools and
scientific and engineering applications that comprised the Encyclopedia of
ComplexityandSystemsSciencearebeingcompletelyupdatedandthemajor-
itywillbepublishedasindividualbookseditedbyexpertsineachfieldwhoare
eminentuniversityfacultymembers.
Thetopicsareasfollows:
AgentBasedModelingandSimulation
ApplicationsofPhysicsandMathematicstoSocialScience
CellularAutomata,MathematicalBasisof
ChaosandComplexityinAstrophysics
ClimateModeling,GlobalWarming,andWeatherPrediction
ComplexNetworksandGraphTheory
ComplexityandNonlinearityinAutonomousRobotics
ComplexityinComputationalChemistry
Complexity in Earthquakes, Tsunamis, and Volcanoes, and Forecasting and
EarlyWarningofTheirHazards
ComputationalandTheoreticalNanoscience
ControlandDynamicalSystems
DataMiningandKnowledgeDiscovery
EcologicalComplexity
ErgodicTheory
FinanceandEconometrics
FractalsandMultifractals
GameTheory
GranularComputing
IntelligentSystems
NonlinearOrdinaryDifferentialEquationsandDynamicalSystems
NonlinearPartialDifferentialEquations
Percolation
PerturbationTheory
ProbabilityandStatisticsinComplexSystems
QuantumInformationScience
SocialNetworkAnalysis
SoftComputing
Solitons
StatisticalandNonlinearPhysics
Synergetics
SystemDynamics
SystemsBiology
Each entry in each of the Series books was selected and peer reviews
organized by one of our university-based book Editors with advice and
SeriesPreface vii
consultationprovidedbyoureminentBoardMembersandtheEditor-in-Chief.
Thislevelofcoordinationassuresthatthereadercanhavealevelofconfidence
in the relevance and accuracy of the information far exceeding than that
generallyfound on theWorld Wide Web. Accessibility isalso a priority and
forthisreasoneachentryincludesaglossaryofimportanttermsandaconcise
definition of the subject. In addition, we are pleased that the mathematical
portions of our Encyclopedia have been selected by Math Reviews for
indexing in MathSciNet. Also, ACM, the world’s largest educational and
scientificcomputingsociety,recognizedourComputationalComplexity:The-
ory,Techniques,andApplicationsbook,whichcontainscontenttakenexclu-
sively from the Encyclopedia of Complexity and Systems Science, with an
awardasoneofthenotableComputerSciencepublications.Clearly,wehave
achievedprominence atalevelbeyond ourexpectations, butconsistentwith
thehighqualityofthecontent!
PalmDesert,CA,USA RobertA.Meyers
September2018 Editor-in-Chief
Volume Preface
Somewherein1930s,whilesippingcoffeewithbrandyinKawiarniaSzkocka
in Lwów, Stanislaw Ulam posed a problem – “Suppose one has an infinite
regularsystemoflatticepointsinEn,eachcapableofexistinginvariousstates
S ,...,S .Eachlatticepointhasawelldefinedsystemofmneighbors,anditis
1 k
assumedthatthestateofeachpointattimet+1isuniquelydeterminedbythe
statesofallitsneighborsattimet.Assumingthatattimetonlyafinitesetof
pointsareactive,onewantstoknowhowtheactivationwillspread.”1Thisis
justoneofthepossibleonsetofcellularautomatatheory.Cellularautomataare
multiorigin and multifarious. They are mathematical machines, models of
computation, architectures of massively parallel processors, and fast pro-
totypingtoolsforstudyingdynamicsofspatiallyextendednonlinearsystems.
AsTommasoToffolitoldmeonce“amagicofcellularautomataisthatthey
havelowentryfeesbuthighexitfees.”Thecellularautomataareverysimple
yet their behavior is often far from predictable, and their analyses require
substantialefforts.Inthisuniquebook,wegatheredacrèmedelacrèmeofthe
cellular automata community. Authors came from different fields of science
anddifferentwalksoflife.Whatmakesthebookuniqueisnotjustsubjectsand
objects of the studies but breadths of cellular automata discoveries made in
mathematics, computers, science, engineering, and physics. I am honored to
thebonestohaveaprivilegeofcompilingthetextsauthoredbybrilliantand
brightestmindsofthescientificandengineeringworld.Thankyou,authors.
Bristol,UK AndrewAdamatzky
September2018 VolumeEditor
1UlamS.M.A.CollectionofMathematicalProblems(NewYork:Interscience,1960),p.30
ix
Cellular Automata Editorial
Acellularautomatonisadiscreteuniversewithdiscretetime,discretespace,
and discrete states. Cells of the universe are arranged into regular structures
calledlatticesorarrays.Eachcelltakesafinitenumberofstatesandupdatesits
states in a discrete time, depending on the states of its neighbors. Cellular
automata aremathematical modelsof massivelyparallel computing; compu-
tationalmodelsofspatiallyextendednonlinearphysical,biological,chemical,
and social systems; and primary tools for studying large-scale complex sys-
tems. Cellular automata are ubiquitous; they are objects of theoretical study
andalsotoolsofappliedmodelinginscienceandengineering.
Commonly, a cellular automaton array is a one- or two-dimensional rect-
angularmatrixofcells.However,othertopologiesarealsoused,e.g.,pentag-
onal tessellations (chapter “▶Cellular Automata in Triangular, Pentagonal,
andHexagonalTessellations,”byCarterBays)andhyperbolicspaces(chapter
“▶Cellular Automata in Hyperbolic Spaces,” by Maurice Margenstern).
Structure of neighborhood, connections between cells, can also change
dynamicallyduringautomaton’sevolution(chapter“▶StructurallyDynamic
Cellular Automata,” by Andrew Ilachinski). Typically, all cells of a cellular
automatonupdatetheirstatessimultaneously,atthesametime,however,there
isafamilyofasynchronousautomatawherecellsmightnothaveaglobalclock
(chapter “▶Asynchronous Cellular Automata,” by Nazim Fates). Cell-state
transitionspersecanbebasedonquantummechanics(chapter“▶Quantum
Cellular Automata,” by Karoline Wiesner). Talking about nonstandard cell-
transition rules, we must mention cellular automata with injective global
functions,whereeveryconfigurationhasexactlyoneprecedingconfiguration
(chapter“▶ReversibleCellularAutomata,”byKenichiMorita).
Typically,acellneighborhoodisfixedduringcellularautomatondevelop-
ment,andacellupdatesitsstatedependingoncurrentstatesofitsneighbors.
Buteveninthisverybasicsetup,thespace-timedynamicsofcellularautomata
are incredibly complex, as can be observed from analysis of the simplest
one-dimensional automata where a transition rule applied to the sum of two
statesisequaltothesumofitsactionsonthetwostatesseparately(seechapter
“▶AdditiveCellularAutomata,”byBurtonVoorhees).Theautomatadynam-
icsbecomesmuchricherifweallowthetopologyofthecellneighborhoodto
beupdateddynamicallyduringautomatondevelopment(seechapter“▶Struc-
turally Dynamic Cellular Automata,” by Andrew Ilachinski) or also allow a
cell’s state to become dependent on the cells’ previous states (see chapter
“▶Cellular Automata with Memory,” by Ramón Alonso-Sanz). Insightful
xi
