# An introduction to Cellular Automata by wanghonghx

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```									An Introduction to
Cellular Automata

Benenson/Torrens (2004)
Chapter 4

GEOG 220 / 2-7-2005
Philipp Schneider
Why CA?

Because they
are great tea
pot warmers…
Overview
   History of CA
   Formal definition of
CA
   Related ideas
   Complex System
Theory and CA
Dynamics
   Urban CA Modeling
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)
from ideas of “comprehensive
modeling” a la Forrester
   In late 1980s, geographers
began to introduce CA ideas in
urban modeling
   Nowadays, CA seem to have a
physics etc. (“Do not mention
Invention of CA
   Invented by John von Neumann
and Stanislaw Ulam at Los
Alamos National Lab (early
1950s)
   Based on work by Alan Turing
   Most basic research on CA in
the 1950s and 60s
   Three major events in CA
research
• John von Neumann’s self-
reproducing automaton
• John Conway’s Game of Life
• Stephen Wolfram’s classification
of cellular automata
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 state
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
certain 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
Neighborhood configurations
   In classic Cellular Automata theory there
are three types of neighborhoods
   Differ in shape and size
   Other configurations have been proposed
but were not accepted
Markov Processes/Fields
   From deterministic to                             p1,1    p1, 2      p1, N
stochastic
   Each cellular automaton can                      p2,1     p 2, 2     p 2, N
be considered as a stochastic        P  pij   
system                                                               
   Transition rules based on                        p N ,1           pN ,N
probabilities
   Similar to CA but transition
rules are substituted by a
matrix of transition
probabilities P

Prob Si  S j   pij

p
j
ij   1

Prob C S i  S j   pij N C 
CA and Complex System Theory
   Game of life         http://www.math.com/studen
   Developed by         ts/wonders/life/life.html
John H. Conway in
1970
   Simple rules 
complex behavior
   Rules
• Survival: 2 or 3
live neighbors
• Birth: exactly 3
live neighbors
• Death: all other
cases
CA Dynamics
   Wolfram’s Classification of
1-D CA behavior
1.   Spatially stable
2.   Sequence of stable or
periodic structures
3.   Chaotic aperiodic behavior
4.   Complicated localized
structures
   Wolframs classification
most popular
   Problem: Class
membership of a given
rule is undecidable
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 simpler
results
   Combination of CA with
differential equations (classical
modeling)
Urban Cellular Automata
   There were a few
in geography in the
1970s but they were
mainly disregarded
   CA matured as a
research tool toward
the end of the 1980s
   Transition began with
raster models that
did not account for
neighborhood
relationships
Raster but not CA
   Raster models possess all characteristics
features
• 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 in
Greensboro, North Carolina
• Buffalo metropolitan area
• Harvard School of Design’s Boston model
Beginning of Urban CA
   Waldo Tobler (1979) took the last step from raster models
to urban CA simulation by introducing a linear transition
function
   Was not accepted by geographic community at first
   Helen Couclelis (1985) recalled Tobler’s work
   CA modeling got accepted by the geographic research
community at the end of the 1980s  many conceptual
papers

g ij t  t   F g ij t , g i 1, j t , g i 1, j t , g i , j 1 t , g i , j 1 t 

            w
  
g
pq i  p , j  q
p1,1 , q 1,1 , p  q 1
t 
Constrained CA
   Extension of original CA idea
   Introduced in 1993 by White
and Engelen (“Constrained
CA model of land-use
dynamics”)
   Mainstream CA application in
geography during the 1990s
   Expansion of the standard
neighborhoods to 113 cells
   Uses the potential of
transition
   Three steps
• Potentials of transition
estimated for each cell
• Obtained potential sorted
decreasingly for each cell
• Externally defined amount of
land distributed over cells
with highest potential
Fuzzy CA models
   Integration of fuzzy set theory
   Based on continuous class membership
functions
   Transition rules describe laws for updating
characteristics based on membership functions

U  x, U x   0,1 | x  X 

      0           for p  p min
 p  pmin

U  p                  for p min  p  pmax
 pmax  pmin

      1           for p max  p
Conclusions
   CA have been around since 1950
   Geography was hesitant to adopt CA as an urban modeling
technique (didn’t 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 distributed
modeling with many applications (urban growth, wildfire