1. Spatio-Temporal Information for Society Münster, 2014
Cellular Automata
Spatio-Temporal Information for Society
Münster, 2014

2. 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.

3. Self-replicating Automata

4. Basic Cellular Automaton
Grid of cells
Neighbourhood
Finite set of discrete states
Finite set of transition rules
Initial state
Discrete time

5. 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.

6. Neighborhood and Rules
Neighbourhood
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.

7.
“CAs contain enough complexity to simulate surprising
and novel change as reflected in emergent phenomena”
(Mike Batty)

8. Source: Rita Zorzenon

9. Game of life

10. 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)

11. Neighborhood
A Neighborhood represents the proximity relations of a cell.
world:createNeighborhood{
strategy = "moore",
self = false }
Von Neumann
Moore

12. 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”} }

13. 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.

14. 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;

15. Which Cellular Automata?
For realistic geographical models
the basic CA principles too constrained to be useful
Extending the basic CA paradigm
From binary (active/inactive) values to a set of inhomogeneous local states
From 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 neighbourhood
From system closure to external events to external output during transitions

16. Game of Life
Static Life
Oscillating Life
Migrating Life

17. Conway’s Game of Life
The universe of the Game of Life is an infinite two-dimensional grid of cells, each of which is either alive or dead. Cells interact with their eight neighbors.

18. Schelling Model for Segregation
Start with a CA with “white” and “black” cells (random)
The new cell state is the state of the majority of the cell’s Moore neighbours
White cells change to black if there are X or more black neighbours
Black cells change to white if there are X or more white neighbours
How long will it take for a stable state to occur?

19. Schelling’s Model of Segregation
Micro-level rules of the game
Stay if at least a third of neighbors are “kin”
< 1/3
Move to random location otherwise

20. One-Dimensional CA’s
Game of Life is 2-D. Many simpler 1-D CAs have been studied
For a given rule-set, and a given starting setup, the deterministic evolution of a CA with one state (on/off) can be pictured as successive lines of colored squares, successive lines under each other

21. Wolfram’s CA classes 1,2
From observation, initially of 1-D CA spacetime patterns, Wolfram noticed 4 different classes of rule-sets. Any particular rule-set falls into one of these:-:
CLASS 1: From any starting setup, pattern converges to all blank -- fixed attractor
CLASS 2: From any start, goes to a limit cycle, repeats same sequence of patterns for ever. -- cyclic attractors

22. Wolfram’s CA classes 3,4
CLASS 3: Turbulent mess, chaos, no patterns to be seen.
CLASS 4: From any start, patterns emerge and
continue continue without repetition for a very long
time (could only be 'forever' in infinite grid)
Classes 1 and 2 are boring, Class 3 is messy,
Class 4 is 'At the Edge of Chaos' - at the transition
between order and chaos -- where Game of Life is!.

23. Wolfram rule 30 current pattern 111 110 101 100 011 010 001 000
new state for center cell 1

24. Wolfram Rule 110

25. Wolfram Rule 110
Proven to be Turing Complete - Rich enough for universal computation
interesting result because Rule 110 is an extremely simple system, simple enough to suggest that naturally occurring physical systems may also be capable of universality

26. Rule 110 Example
Requires potentially infinite dimensions for general computation

27. Classifying Cellular Automata Rules
Class One - Fixed or Static: Rules that produce dull universes, such as all dead cells or all living cells; e.g. ice.
Class Two - Periodic or Oscillatory: Rules that produce stable, repetitive configurations.
Class Three - Chaotic: Rules that produce chaotic patterns; e.g. molecules in a gas.
Class Four - Complexity: Rules that produce complex, locally organized patterns; e.g. behave like a flowing liquid..

28. Celular automata classifications
 = chance that a cell is alive in the next state