Cellular Automata

Pedro R. Andrade

Münster, 2013

System Theory

AdvantagesSimple representation of the worldVisual representationModular and hierarchical

DisadvantagesNo heterogeneityImplicit spatial representationFixed connections between stocks

Cellular Automata

Firstly developed by Hungarian mathematician John von Neumann, who proposed a model based on the idea of logical systems that were self-replicating.

Self-replicating Automata

Basic Cellular Automaton

Grid of cells Neighbourhood Finite set of discrete states Finite set of transition rules Initial state Discrete time

2-Dimensional Automaton

A 2-dimensional cellular automaton consists of an infinite (or finite) grid of cells, each in one of a finite number of states. Time is discrete and the state of a cell at time t is a function of the states of its neighbors at time t-1.

Neighborhood and Rules

RulesNeighbourhood

States

Space and Time

t

t1

Each cell is autonomous and change its state according to its current state and the state of its neighborhood.

www.terrame.org

“CAs contain enough complexity to simulate surprising and novel change as reflected in emergent phenomena”(Mike Batty)

9

Source: Rita Zorzenon

Game of life

CellularSpace

A CellularSpace is a set of Cells. It consists of an area of interest, divided into a regular grid.

world = CellularSpace{xdim = 5,ydim = 5

}

forEachCell(world, function(cell)cell.value = 3

end)

Neighborhood A Neighborhood represents the proximity relations

of a cell.

world:createNeighborhood{

strategy = "moore",self = false

}

Von Neumann Moore

Legend

Defines colors to draw the Cells of a CellularSpace. Can be used with map observers.

coverLeg = Legend {grouping = "uniquevalue",colorBar = {

{value = 0, color = "white"},{value = 1, color = "red"},{value = 2, color = "green”}

}}

Synchronizing a CellularSpace

TerraME can keep two copies of a CellularSpace in memory: one stores the past values of the cells, and another stores the current (present) values of the cells.

The model equations must read the past copy and write the values to the present copy of the cellular space.

At the correct moment, it will be necessary to synchronize the past copy with the current values of the cellular space.

Characteristics of CA models

Self-organising systems with emergent properties: locally defined rules resulting in macroscopic ordered structures. Massive amounts of individual actions result in the spatial structures that we know and recognise;

Which Cellular Automata?

For realistic geographical modelsthe basic CA principles too constrained to be useful

Extending the basic CA paradigm From binary (active/inactive) values to a set of

inhomogeneous local statesFrom discrete to continuous values (30% cultivated land, 40%

grassland and 30% forest)Transition rules: diverse combinations Neighborhood definitions from a stationary 8-cell to

generalized neighbourhoodFrom system closure to external events to external output

during transitions