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)
   CA paradigm needed departure
    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
    bad reputation in mathematics,
    physics etc. (“Do not mention
    CA in your CV!”)
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
    publications about CA
    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
    spread, transportation)

								
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