1. An Introduction to Cellular Automata
Benenson/Torrens (2004) Chapter 4
GEOG 220 /
Philipp Schneider

2. Why CA?
Because they are great tea pot warmers…

3. Overview History of CA Formal definition of CA Related ideas
Complex System Theory and CA Dynamics
Urban CA Modeling

4. 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!”)

5. 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

6. CA Definition General Formal definition
“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

7. 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

8. Markov Processes/Fields
From deterministic to stochastic
Each cellular automaton can be considered as a stochastic system
Transition rules based on probabilities
Similar to CA but transition rules are substituted by a matrix of transition probabilities P

9. CA and Complex System Theory
Game of life
Developed by 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

10. CA Dynamics Wolfram’s Classification of 1-D CA behavior
Spatially stable
Sequence of stable or periodic structures
Chaotic aperiodic behavior
Complicated localized structures
Wolframs classification most popular
Problem: Class membership of a given rule is undecidable

11. 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)

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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)