Philipp CA

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.html


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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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