An Introduction to Cellular Automata

1. An Introduction to Cellular Automata Benenson/Torrens (2004) Chapter 4 GEOG 220 / 2-7-2005 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 • “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 http://www.math.com/students/wonders/life/life.html

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)