An Introduction to An Introduction to Cellular AutomataCellular Automata

Benenson/Torrens (2004) Benenson/Torrens (2004) Chapter 4Chapter 4

GEOG 220 / 2-7-2005GEOG 220 / 2-7-2005Philipp SchneiderPhilipp Schneider

Why CA?Why CA?

Because they are great tea pot warmers…

OverviewOverview

History of CAHistory of CA Formal definition of Formal definition of

CACA Related ideasRelated ideas Complex System Complex System

Theory and CA Theory and CA DynamicsDynamics

Urban CA ModelingUrban CA Modeling

History of Urban CA modelsHistory of Urban CA models Based on two ideasBased on two ideas

• Raster conceptualization of space Raster conceptualization of space (late 1950s)(late 1950s)

• Regional modeling of flows of Regional modeling of flows of population, goods, jobs etc. (1960s population, goods, jobs etc. (1960s and 1970s)and 1970s)

CA paradigm needed departure CA paradigm needed departure from ideas of “comprehensive from ideas of “comprehensive modeling” a la Forrestermodeling” a la Forrester

In late 1980s, geographers In late 1980s, geographers began to introduce CA ideas in began to introduce CA ideas in urban modelingurban modeling

Nowadays, CA seem to have a Nowadays, CA seem to have a bad reputation in mathematics, bad reputation in mathematics, physics etc. (“Do not mention physics etc. (“Do not mention CA in your CV!”)CA in your CV!”)

Invention of CAInvention of CA Invented by John von Neumann Invented by John von Neumann

and Stanislaw Ulam at Los and Stanislaw Ulam at Los Alamos National Lab (early Alamos National Lab (early 1950s)1950s)

Based on work by Alan TuringBased on work by Alan Turing Most basic research on CA in Most basic research on CA in

the 1950s and 60sthe 1950s and 60s Three major events in CA Three major events in CA

researchresearch• John von Neumann’s self-John von Neumann’s self-

reproducing automatonreproducing automaton• John Conway’s Game of LifeJohn Conway’s Game of Life• Stephen Wolfram’s Stephen Wolfram’s

classification of cellular classification of cellular automataautomata

CA DefinitionCA Definition GeneralGeneral

• ““A system made up of many discrete cells, each of which may be in A system made up of many discrete cells, each of which may be in one of a finite number of states. A cell or automaton may change state one of a finite number of states. A cell or automaton may change state only at fixed, regular intervals, and only in accordance with fixed rules only at fixed, regular intervals, and only in accordance with fixed rules that depend on cells own values and the values of neighbors within a that depend on cells own values and the values of neighbors within a certain proximity. “certain proximity. “

Formal definitionFormal definition• CA = one- or two-dimensional grid of identical automata cellsCA = one- or two-dimensional grid of identical automata cells• Each cell processes information and proceeds in its actions Each cell processes information and proceeds in its actions

depending on its neighborsdepending on its neighbors• Each cell (automaton) A defined by Each cell (automaton) A defined by

Set of States S = {SSet of States S = {S11, S, S22, S, S33, …, S, …, SNN}} Transition Rules TTransition Rules T

• Therefore A ~ (S,T,R)Therefore A ~ (S,T,R) (R: neighboring automata)(R: neighboring automata)• T: (ST: (Stt, I, Itt) ) S St+1t+1

Neighborhood configurationsNeighborhood configurations In classic Cellular Automata theory there In classic Cellular Automata theory there

are three types of neighborhoodsare three types of neighborhoods Differ in shape and sizeDiffer in shape and size Other configurations have been proposed Other configurations have been proposed

but were not acceptedbut were not accepted

Markov Processes/FieldsMarkov Processes/Fields From deterministic to From deterministic to

stochasticstochastic Each cellular automaton can Each cellular automaton can

be considered as a stochastic be considered as a stochastic systemsystem

Transition rules based on Transition rules based on probabilitiesprobabilities

Similar to CA but transition Similar to CA but transition rules are substituted by a rules are substituted by a matrix of transition matrix of transition probabilities Pprobabilities P

NNN

N

N

ij

pp

ppp

ppp

p

,1,

,22,21,2

,12,11,1

P

ijji pSS Prob

j

ijp 1

CNpSS ijji CProb

CA and Complex System TheoryCA and Complex System Theory

Game of lifeGame of life Developed by John Developed by John

H. Conway in 1970H. Conway in 1970 Simple rules Simple rules

complex behaviorcomplex behavior RulesRules

• Survival: 2 or 3 live Survival: 2 or 3 live neighborsneighbors

• Birth: exactly 3 live Birth: exactly 3 live neighborsneighbors

• Death: all other Death: all other casescases

http://www.math.com/students/wonders/life/life.html

CA DynamicsCA Dynamics Wolfram’s Classification of Wolfram’s Classification of

1-D CA behavior1-D CA behavior1.1. Spatially stableSpatially stable2.2. Sequence of stable or Sequence of stable or

periodic structuresperiodic structures3.3. Chaotic aperiodic behaviorChaotic aperiodic behavior4.4. Complicated localized Complicated localized

structuresstructures Wolframs classification Wolframs classification

most popularmost popular Problem: Class Problem: Class

membership of a given membership of a given rule is undecidablerule is undecidable

Variations of Classic CAVariations of Classic CA Grid geometry & NeighborhoodGrid geometry & Neighborhood

• Hexagonal, triangular and Hexagonal, triangular and irregular gridsirregular grids

• Larger or more complicated Larger or more complicated neighborhoodsneighborhoods

• generally do not introduce any generally do not introduce any significant effectsignificant effect

Synchronous and Synchronous and asynchronous CAasynchronous CA• Sequential update Sequential update • Parallel updateParallel update• In general, asynchronously In general, asynchronously

updated CA produce simpler updated CA produce simpler resultsresults

Combination of CA with Combination of CA with differential equations (classical differential equations (classical modeling)modeling)

Urban Cellular AutomataUrban Cellular Automata There were a few There were a few

publications about CA publications about CA in geography in the in geography in the 1970s but they were 1970s but they were mainly disregardedmainly disregarded

CA matured as a CA matured as a research tool toward research tool toward the end of the 1980sthe end of the 1980s

Transition began with Transition began with raster models that raster models that did not account for did not account for neighborhood neighborhood relationshipsrelationships

Raster but not CARaster but not CA Raster models possess all characteristics Raster models possess all characteristics

featuresfeatures• Use of cellular spaceUse of cellular space• Cells characterized by stateCells characterized by state• Models are dynamicModels are dynamic

BUT: They lack dependence of cell state BUT: They lack dependence of cell state on states of neighboring cellson states of neighboring cells

Examples Examples • Simulation of urban development in Simulation of urban development in

Greensboro, North CarolinaGreensboro, North Carolina• Buffalo metropolitan areaBuffalo metropolitan area• Harvard School of Design’s Boston modelHarvard School of Design’s Boston model

Beginning of Urban CABeginning of Urban CA Waldo Tobler (1979) took the last step from raster models Waldo Tobler (1979) took the last step from raster models

to urban CA simulation by introducing a linear transition to urban CA simulation by introducing a linear transition functionfunction

Was not accepted by geographic community at firstWas not accepted by geographic community at first Helen Couclelis (1985) recalled Tobler’s workHelen Couclelis (1985) recalled Tobler’s work CA modeling got accepted by the geographic research CA modeling got accepted by the geographic research

community at the end of the 1980s community at the end of the 1980s many conceptual many conceptual paperspapers

1,1,1,1,1

,

1,1,,1,1 ,,,,

qpqp

qjpipq

jijijijiijij

tgw

tgtgtgtgtgFttg

Constrained CAConstrained CA Extension of original CA ideaExtension of original CA idea Introduced in 1993 by White Introduced in 1993 by White

and Engelen (“Constrained and Engelen (“Constrained CA model of land-use CA model of land-use dynamics”)dynamics”)

Mainstream CA application in Mainstream CA application in geography during the 1990sgeography during the 1990s

Expansion of the standard Expansion of the standard neighborhoods to 113 cellsneighborhoods to 113 cells

Uses the potential of Uses the potential of transition transition

Three stepsThree steps• Potentials of transition Potentials of transition

estimated for each cellestimated for each cell• Obtained potential sorted Obtained potential sorted

decreasingly for each celldecreasingly for each cell• Externally defined amount of Externally defined amount of

land distributed over cells land distributed over cells with highest potentialwith highest potential

Fuzzy CA modelsFuzzy CA models

Integration of fuzzy set theoryIntegration of fuzzy set theory Based on continuous class membership Based on continuous class membership

functionsfunctions Transition rules describe laws for updating Transition rules describe laws for updating

characteristics based on membership functionscharacteristics based on membership functions

XxxxU U |1,0,

pfor p

ppfor ppp

pp pfor p

pU

max

maxminminmax

min

min

1

0

ConclusionsConclusions CA have been around since 1950CA have been around since 1950 Geography was hesitant to adopt CA as an urban modeling Geography was hesitant to adopt CA as an urban modeling

technique (didn’t happen before the mid-1980stechnique (didn’t happen before the mid-1980s Since then, many extensions of CA have been proposed, Since then, many extensions of CA have been proposed,

some effective, others notsome effective, others not Nowadays CA are a valuable tool for spatially distributed Nowadays CA are a valuable tool for spatially distributed

modeling with many applications (urban growth, wildfire modeling with many applications (urban growth, wildfire spread, transportation)spread, transportation)