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Cellular Automata

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A cellular automaton is a discrete model studied in computer science, mathematics, physics, complexity science, theoretical biology and microstructure modeling.

This presentation is a basic introduction.

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Cellular Automata

  1. 1. CPAD|KirthiBalakrishnan|SimrithiSKumar
  2. 2. A cellular automaton consists of a regular grid of cells, each in one of a finite number of states, such as on and off (in contrast to a coupled map lattice). The grid can be in any finite number of dimensions. For each cell, a set of cells called its neighborhood is defined relative to the specified cell. An initial state (time t = 0) is selected by assigning a state for each cell. A new generation is created (advancing t by 1), according to some fixed rule (generally, a mathematical function)[3] that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood. What is ? 2 CA in a dodeca grid
  3. 3. To illustrate how CA works, we first define ● a grid of cells, ( or it could be irregular but to simplify we will assume a square grid) ● a neighbourhood around each cell which is composed of the nearest cells, ● a set of rules as to how what happens in the neighbourhood affects the development of the cell in question ● a set of states that each cell can take on – i.e. developed or not developed ● an assumption of universality that all these features operate uniformly and universally
  4. 4. Each box stands for a student wearing (black) or not wearing (white) a hat. Let us make the two following assumptions: ● Hat rule: a student will wear the hat in the following class if one or the other—but not both—of the two classmates sitting immediately on her left and on her right has the hat in the current class (if nobody wears a hat, a hat is out of fashion; but if both neighbors wear it, a hat is now too popular to be trendy). ● Initial class: during the first class in the morning, only one student in the middle shows up with a hat.
  5. 5. 5 Consecutive rows represent the evolution in time through subsequent classes. The evolutionary pattern displayed contrasts with the simplicity of the underlying law (the “Hat rule”) and ontology (for in terms of object and properties, we only need to take into account simple cells and two states). The global, emergent behavior of the system supervenes upon its local, simple features, at least in the following sense: the scale at which the decision to wear the hat is made (immediate neighbors) is not the scale at which the interesting patterns become manifest.
  6. 6. Even perfect knowledge of individual decision rules does not always allow us to predict macroscopic structure. We get macro-surprises despite complete micro-knowledge. ” —Epstein (1999: 48)
  7. 7. There’s an amazing diversity of forms. And, yes, they’re often complicated. But because they’re based on simple underlying rules, they always have a certain logic to them: in a sense each of them tells a definite “algorithmic story”.
  8. 8. ● Essentially CA models developed in the late 1980s early 1990s from at least three sources: bottom up thinking about systems in contrast to top down, concepts of emergence in particular related to morphology, GIS and raster based representation of activity layers. ● These models have found favour in rapidly growing systems which are characterised by urban sprawl, like Phoenix. They have been quite inappropriately applied to non ‐ rapid growth cities where the focus is on redistribution. ● They have not been widely applied by municipalities as they do not contain explicit mechanisms for generating numerical forecasts that are demographically or economically based.
  9. 9. (a) The neighbourhood is composed of 8 cells around the central cell How a CA works defined on a grid of cells with two states – not developed & developed 9 (b) Place the neighbourhood over each cell on the grid. The rule says that if there is one or more cells developed (black) in the neighbourhood, then the cell is developed. (c) If you keep on doing this for every cell, you get the diffusion from the central cell shown below.
  10. 10. MOORE VON NEUMANN EXTENDED MOOR VON NEUMANN composed of different combinations of cells in strictly deterministic CA models
  11. 11. For example, for any cell {x,y}, ● if only one neighborhood cell either NW, SE, NE, or SW other than {x,y} is already developed, ● then cell {x,y} is developed according to the following neighborhood switching rule
  12. 12. For probabilistic rules, we can generate statistically self‐similar structures which look more like real city morphologies. For example, ● if any neighborhood cell other than {x,y} is already developed, then the field value p {x,y} is set ● & if p {x,y} > some threshold value, then the cell {x,y} is developed
  13. 13. Over and over again we will see the same kind of thing: that even though the underlying rules for a system are simple, and even though the system is started from simple initial conditions, the behavior that the system shows can nevertheless be highly complex. ” —Wolfram (2002: 28)
  14. 14. 15 Any questions? SOURCES: ● Stanford Plato Entry: Cellular Automata ● Steve Wolfram: Rule 30 in CA ● CA Lecture Notes: UCL