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8/12/2019 Philipp CA
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An Introduction to
Cellular Automata
Benenson/Torrens (2004)Chapter 4
GEOG 220 / 2-7-2005
Philipp Schneider
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Why CA?
Because theyare great teapot warmers
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Overview
History of CA
Formal definition ofCA
Related ideas
Complex System
Theory and CADynamics
Urban CA Modeling
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History of Urban CA models
Based on two ideas Raster conceptualization of space
(late 1950s) Regional modeling of flows of
population, goods, jobs etc. (1960s
and 1970s) CA paradigm needed departure
from ideas of comprehensivemodeling a la Forrester
In late 1980s, geographers
began to introduce CA ideas inurban modeling Nowadays, CA seem to have a
bad reputation in mathematics,physics etc. (Do not mention
CA in your CV!)
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Invention of CA
Invented by John von Neumannand Stanislaw Ulam at LosAlamos National Lab (early1950s)
Based on work by Alan Turing Most basic research on CA in
the 1950s and 60s
Three major events in CAresearch
John von Neumanns self-reproducing automaton
John Conways Game of Life
Stephen Wolframs classificationof cellular automata
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CA Definition
General 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 stateonly at fixed, regular intervals, and only in accordance with fixed rulesthat depend on cells own values and the values of neighbors within acertain proximity.
Formal definition CA = one- or two-dimensional grid of identical automata cells Each cell processes information and proceeds in its actions
depending on its neighbors Each cell (automaton) A defined by
Set of States S = {S1, S2, S3, , SN} Transition Rules T
Therefore A ~ (S,T,R) (R: neighboring automata) T: (St, It) St+1
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Neighborhood configurations
In classic Cellular Automata theory thereare three types of neighborhoods
Differ in shape and size Other configurations have been proposed
but were not accepted
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Markov Processes/Fields
From deterministic tostochastic
Each cellular automaton canbe considered as a stochasticsystem
Transition rules based on
probabilities Similar to CA but transition
rules are substituted by amatrix of transitionprobabilities 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
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CA and Complex System Theory
Game of life
Developed byJohn H. Conway in1970
Simple rules complex behavior
Rules Survival: 2 or 3
live neighbors
Birth: exactly 3live neighbors
Death: all othercases
http://www.math.com/students/wonders/life/life.html
http://www.math.com/students/wonders/life/life.htmlhttp://www.math.com/students/wonders/life/life.htmlhttp://www.math.com/students/wonders/life/life.htmlhttp://www.math.com/students/wonders/life/life.html8/12/2019 Philipp CA
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CA Dynamics
Wolframs Classification of1-D CA behavior
1. Spatially stable
2. Sequence of stable orperiodic structures
3. Chaotic aperiodic behavior
4. Complicated localizedstructures
Wolframs classificationmost popular
Problem: Classmembership of a givenrule is undecidable
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Variations of Classic CA
Grid geometry & Neighborhood Hexagonal, triangular and
irregular grids Larger or more complicated
neighborhoods generally do not introduce any
significant effect Synchronous and
asynchronous CA Sequential update Parallel update In general, asynchronously
updated CA produce simplerresults
Combination of CA withdifferential equations (classicalmodeling)
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Urban Cellular Automata
There were a fewpublications about CAin geography in the1970s but they were
mainly disregarded CA matured as a
research tool towardthe end of the 1980s
Transition began with
raster models thatdid not account forneighborhoodrelationships
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Raster but not CA
Raster models possess all characteristicsfeatures Use of cellular space Cells characterized by state
Models are dynamic BUT: They lack dependence of cell state
on states of neighboring cells Examples
Simulation of urban development inGreensboro, North Carolina Buffalo metropolitan area Harvard School of Designs Boston model
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Beginning of Urban CA
Waldo Tobler (1979) took the last step from raster modelsto urban CA simulation by introducing a linear transitionfunction
Was not accepted by geographic community at first
Helen Couclelis (1985) recalled Toblers work
CA modeling got accepted by the geographic researchcommunity at the end of the 1980s many conceptualpapers
1,1,1,1,1
,
1,1,,1,1 ,,,,
qpqp
qjpipq
jijijijiijij
tgw
tgtgtgtgtgFttg
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Constrained CA
Extension of original CA idea Introduced in 1993 by White
and Engelen (ConstrainedCA model of land-usedynamics)
Mainstream CA application ingeography during the 1990s
Expansion of the standardneighborhoods to 113 cells
Uses the potential oftransition
Three steps Potentials of transitionestimated for each cell
Obtained potential sorteddecreasingly for each cell
Externally defined amount ofland distributed over cells
with highest potential
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Fuzzy CA models
Integration of fuzzy set theory
Based on continuous class membershipfunctions
Transition rules describe laws for updatingcharacteristics based on membership functions
XxxxU U |1,0,
pfor p
ppfor ppp
pp
pfor p
pU
max
maxmin
minmax
min
min
1
0
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Conclusions
CA have been around since 1950 Geography was hesitant to adopt CA as an urban modeling
technique (didnt happen before the mid-1980s Since then, many extensions of CA have been proposed,
some effective, others not
Nowadays CA are a valuable tool for spatially distributedmodeling with many applications (urban growth, wildfirespread, transportation)
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