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

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 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.

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

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 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

Game of Life Static Life Oscillating Life Migrating Life

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.

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?

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

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

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

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!.

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

Wolfram Rule 110

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

Rule 110 Example Requires potentially infinite dimensions for general computation

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..

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