ca.nlogo

;===========================================================
; CA: A framework for cellular-automata
; by Jon Pearce (www.cs.sjsu.edu/faculty/pearce/pearce.html)s
;===========================================================

;==================================
;=====      Declarations      =====
;==================================

patches-own [ 
   state      
]

globals [ ]

;==================================
;===== Initializing the Model =====
;==================================
to init-model
   ca ; clear all
   random-seed new-seed ; randomly seed random number generator
   init-globals
   init-patches
end

to init-patches
   ask patches [init-patch]
end

;=====Initialization overridables =====

to init-globals

end

to init-patch
   set state random num-states
   color-patch
end

;==================================
;=====   Updating the Model   =====
;==================================

to update-model
   if finished?
   [ 
      print "Simulation halted"
      stop 
   ]
   tick ; increment the tick counter
   update-globals
   update-patches
   
end

to update-patches
   ask patches [update-patch]
end

; ===== Update overridables =====

to update-globals
   ; To do: update values of globals.
end

to-report finished?
   ; To do: report model halting condition.
   report false ; for now
end

to update-patch
   let my-neighbors other patches with [distance myself <= radius]
   let nbhd-state [state] of my-neighbors
   set state one-of modes nbhd-state
   color-patch
end

to color-patch
   let macro-state-size ceiling (num-states / 14)       
   let macro-state ceiling (state / macro-state-size)       
   let named-color 10 * macro-state + 5       
   set pcolor named-color     
end



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@#$#@#$#@
WHAT IS IT?
-----------
CA is a framework for creating 2-dimensional cellular automata models (2D-CA) in NetLogo.

HOW IT WORKS
------------

An NetLogo 2D-CA model consists of a two-dimensional array of cells (patches). Each cell has a state, which can be any value taken from some state space, and a neighborhood. The neighborhood of a cell, c, are all of those cells n such that:

|   n != c
|   distance c n <= radius

In otherwords, n is in c's neighborhood, or n is a neighbor of c, if the shortest path from c to n traverses at most radius cells.

For simplicity, the state spcace can be taken as all integers in the interval:

|   [0, num-states)

In addition, a 2D-CA has:

|   1. a procedure for initializing the state of a cell: init-patch
|   2. a procedure for updating the state of a cell: update-patch
|   3. a test that determines if the simulation should halt (finished?)

A simulation (i.e., running the model) is a run of the following control loop:

|   init-model
|   while [not finished?] [ update-model ]

The init-model and update-model procedures simply ask each cell to execute the init-patch and update-patch procedures, respectively.

To update a patch, c, the states of all of c's neighbors are collected into a set. Next, the neighborhood state is computed. This is the list of states of every patch in c's neighborhood. The built-in modes function reports a list of the most commonly occuring states in the neighborhood state. For example, if the neighborhood state is:

|   [1 2 3 4 2 3 2 3]

Then the modes will be the list:

|  [2 3]

One of these commonly occuring states is chosed at random using the built-in one-of function. Finally, the patch is colored. Here's the complete code:

|   to update-patch
|      let my-neighbors other patches with [distance myself <= radius]
|      let nbhd-state [state] of my-neighbors
|      set state one-of modes nbhd-state
|      color-patch
|   end

The update-patch procedure is overridable. Possible implementations include averaging the neighbor's states, choosing the maximum or minimum neighbor state, etc.

For example:

|   to update-patch
|      let my-neighbors other patches with [distance myself <= radius]
|      let nbhd-state [state] of my-neighbors
|      set state mean nbhd-state
|      color-patch
|   end

MAPPING STATE TO COLOR
-------------------

What is the ideal mapping of the state space, [0, num-states), into the NetLogo color space, [0, 140)?  If the state space is large, then different states can map to indistinguishable colors, "close" states can map to "distant" colors, and distant states can map to close colors.

To partially remedy this problem, we observe that the named colors in the NetLogo color space (excluding black and white) are encoded by the numbers 5, 15, 25, ..., 135. Let's call the set of these numbers the named-color-space:

|   named-color-space = {5, 15, 25, ..., 135}

Note that there are 14 members in the named-color-space. 

We can partition the state space into 14 macro states by integer division:

|   set macro-state state / macro-state-size

where 

|   macro-state-size = num-states / 14

We map the macro-state onto a named color as follows:

|   set named-color 10 * macro-state + 5

Here's the complete procedure:

|   to color-patch
|      let macro-state-size ceiling (num-states / 14)
|      let macro-state ceiling (state / macro-state-size)
|      let named-color 10 * macro-state + 5
|      set pcolor named-color
|    end

For example, if num-states is 70, then macro-state-size will be 70/14 = 5. We can think of the macro states as the 14 length 5 intervals:
  
|   [0, 5), [5, 10), [10, 15), ..., [65, 70)

If the state of a given cell, c, is 50, then its macro-state will be 50/5 = 10. This corresponds to the 10th interval: [50, 55). In this case the color of c will be 10 * 10 + 5 = 105 = blue. 

If the state of c is 63, then its macro-state will be 12 (= ceiling of 12.6) and its color will be 125 = magenta.

What colors do states 0 and 69 map onto?

As another example, if num-states is 14, then macro-state-size will be 14/14 = 1. If the state of cell c is 12, then its micro-state will also be 12 and its color will be 125. 

If num-states is 2, then macro-state-size will again be ceiling 2/14 = 1. If the state of cell c is 1, then its color will be 15 (red) otherwise its color will be 5 (grey).

With this in place we can add a slider to allow the user to adjust num-states.

HOW TO USE IT
-------------

If the control loop (described above) is running, pressing the UPDATE button pauses it. Otherwise the control-loop starts or resumes.

Pressing the INIT button initializes the 2D-CA.

The radius slider allows the user to experiment with different values for the radius variable described above.

THINGS TO NOTICE
----------------

Notice the evolving color patterns as the simulation runs. 

Do all of the patches eventually have the same color or are their "islands" of patches with a different color from their surroundings? How could this happen?

HOW MANY UPDATE-PATCH PROCEDURES ARE THERE?
-------------------
It's easy to see that the number of possible neighborhood states is: 

|   num-nbhd-states = num-states ^ (length nbrhd-state)

Our update-patch procedure maps each of these neighborhood states into a state. This means the number of update procedures is:

|   num-update-rules = num-states ^ num-nbrhd-states

We can therefore list all of these update procedures:

|   update-0, update-1, ..., update-k

where

|   k = num-update-rules - 1

For example, if radius = 1, then cell c has 8 neighbors. If num-states = 2, then there are 2^8 = 256 neighborhood states:

|   (0, 0, 0, ..., 0)
|   (0, 0, 0, ..., 1)
|   ...
|   (1, 1, 1, ..., 1)

This means there are 2 ^ 256 possible update rules. This is a huge number!

AGGREGATE STATE AND COMPLEXITY
-------------------
The aggregate state of the model, or the model state, is simply the list of states of all of the cells in the model:

|   set model-state [state] of patches

The update-patch procedure can be arbitrarily complex, but it doesn't need to be in order to generate complex or interesting patterns of change in the model-state. In such cases it might be difficult to predict patterns in the model state by studying how individual cells update themselves. 

The system (model) is more than the sum of its parts (cells). It has its own behavioral patterns that can't be explained by the behavior of its parts. It creates the impression that the system has an identity distinct from its identity as a collection of parts. Many natural and social systems have this property. Examples include societies, economies, ecosystems, brains, and corporations. Such systems are difficult to study using top-down or reductionist methods.

APPLICATIONS TO SOCIOLOGY
-------------------
The economist Thomas Schelling first applied this idea to sociology when he speculated that people who lived in segragated neighborhoods weren't necessarily racists. In this case we might think of patches as homes. The state of a patch is the race of the occupant. The update-patch procedure can be interpreted to say that the occupant of a home simply wants the majority of his neighbors to be of the same race. In other words, the occupant will tolerate neighbors of different races. Never the less, when we randomly assign occupants to homes, then run the simulation, we notice that segragated neighborhoods develop. 

Instead of race, the state of a patch can be the political party affiliation of the occupant. In this case the update-patch procedure can be interpreted as expressing the truism that we vote like our neighbors vote (despite the feeling that our political opinions are purely rational.) Perhaps this explains the red-state-blue-state phenomenon in US presidential elections. 

See http://www.theatlantic.com/doc/200204/rauch for more background on this.


PROJECT: MODELING CORRUPTION
-------------------
In the corruption model patches are individuals. The state of a patch is one of the strings: "corrupt", "honest", or "jailed". Initially no one is in jail, but a slider allows the user to control the initial number of corrupt individuals.

The update-patches procedure has two phases. In phase one it asks all unjailed patches to interact with their neighbors. In phase two it asks all patches to update themselves.

During the interaction phase each unjailed patch, p, selects a random set of its unjailed neighbors. For each such neighbor, n, if p is honest but n is corrupt, then p files a complaint against n. If n is honest but p is corrupt, then n files a complaint against p.

During the update phase each unjailed patch, p, computes the percentage of its neighbors that are jailed. If this ratio is below the crime-pays threshold, then p will change its state from honest to corrupt. If this ratio is above the too-risky threshold, then p will change its state from corrupt to honest. However, if p is corrupt, and the number of complaints against p exceeds the crime-tolerance threshold, then p will change its state from corrupt to jailed.

If p is a jailed patch, an if the number of ticks since p was jailed exceeds the jail-time, then p changes its state from jailed to honest.

Provide users with sliders to control the initial number of corrupt patches, the crime-pays threshold, the too-risky threshold, the crime-tolerance threshold, and jail-time. Also provide a plotter that plots the percentage of patches that are honest, corrupt, or jailed at tick t.

What can this model tell policy makers about the effectiveness of increasing sentences versus putting more police on the street? 

We can improve the model by giving each patch an influence attribute. For most patches the value of this attribute is 1, but for some the value can be higher. When an unjailed patch updates its state, it computes three ratios: honest-neighbors, corrupt-neighbors, and jailed-neighbors. Influential neighbors have greater weight in the computation of each of these ratios. The new state of p is based on some sensible combination of these ratios. For example, if most of p's neighbors are corrupt, and if few are in jail, then p becomes corrupt. If most are honest and if many are in jail, the p becomes honest. etc. We this in place we can study the policy of targeting influential individuals.

This model is based on a 1D-CA model developed by Ross Hammond. The model is dicussed in the article posted at http://www.theatlantic.com/doc/200204/rauch

PROJECT: 1D-CA
-------------------

As we have seen, the number of update procedures in a 2D-CA is huge. We can reduce the number of rules without losing expressiveness by considering one-dimensional cellular automata (1D-CA). These models were shown by Wolfram [Wolf] to be universal in the sense that any Turing machine can be encoded by the computation generated by some 1D-CA. 

A 1D-CA is a row of cells. Assume each cell can be in one of two states: 0 or 1. Each cell has two neighbors, three including itself. Therefore there are 8 possible neighborhood states. Each of these states can map to one of two new states by the update function. Therefore there are 2^8 = 256 possible update rules. 

How can an update rule be encoded as an integer between 0 and 256?

Assume the cells in row i of the NetLogo world represent some 1D-CA at time i. Since the default world consists of 32 x 32 cells, we can model 32 updates of a 1D-CA consisting of 32 cells. Is it possible to "wrap" time?

Provide a control that will allow the user to specify one of 256 rules.

Hint: 

You might want to define a global called world that is simply a shuffled list of 32 bits (i.e., 0's and 1's). 

Also define globals called clock and row with initial values 0 and 32, respectively.

The update-globals procedure updates world, clock, and row. Clock is incremented. World is updated according to the selected 1D-CA update rule. 

If the value of clock is divisible by k, for some suitable value of k (e.g., k = 10), then row is decremented and (patch i row) is colored red or blue according to the value of (item i world).

PROJECT: DISSEMINATING CULTURE
-------------------
In this project we use a NetLogo 2D-CA to study the dissemination of culture. The model is based on Robert Axelrod's paper: Disseminating Culture, which can be found in [Axelrod].

Assume each patch represents an ethnic region. Assume the state of a patch, called its culture, is a list consisting of N cultural features. For example:

|   position 0 = religion
|   position 1 = technology
|   position 2 = political organization
|   position 3 = economic system
|   position 4 = language
|   etc.

Assume the value of a cultural feature, called a trait, is an integer, t such that:

|   0 <= t < M

For example, the trait at position 0 might indicate the type of religion:

|   0 = Animism
|   1 = Hinduism
|   2 = Buddhism
|   3 = Christianity
|   4 = Judaism
|   5 = Islam
|   etc.

How many cultures are there in our model?

Let's assume the absolute value of the difference between two traits corresponds to their cultural distance. For example, if stone age technology is 0 then information age technology might be 8 indicating that the difference is very large. (Of course not all traits can be ordered in a linear way.) How can the color of a patch reflect its state in such a way that similar cultures have similar colors and dissimilar cultures have noticably different colors?

Initially the state of each patch is random.

To update the model:

|   1. For each patch, p1, pick a random neighbor, p2. 
|   2. Compute s = the percentage of features that p1 and p2 have in common.
|   3. Pick a random number n < 100. If n < s, then p1 borrows a trait from p2

Ideally, p1 borrows a trait from p2 other than one they already have in common.

Hint: In the RGB color space there are 256^3 = 2^24 colors. Since 24 = 6 * 4, this suggests we can take N = 4 and M = 6. Of course there are only 140 colors in the NetLogo color space. This might suggest choosing N = 5 and M = 3.

PROJECT: FOREST FIRE
-------------
Imagine each patch is a patch of ground in a forest. The state of a patch has three possible values: "has-tree" (pcolor = green), "burned" (pcolor = black), and "empty" (pcolor = brown). A patch also has a fertility attribute, which is a number between 0 and 1. Initially the state of a patch is empty and its fertility is set to a constant, init-fertility, which the user can adjust with a slider.

Every spring (cycles = an even number), each patch grows a tree with probability = fertility. Each summer (cycles = an odd number) lightening strikes a patch with probability = burn-probability, which the user can control with a slider.

If a patch has a tree and is struck by lightening, then its state changes to "burned". The fire spreads: all neighboring patches with trees burn. Their treed neighbors burn, and so on until the fire reaches a fire break of some sort. 

A plotter plots the percentage of the patches that are in the have-tree state.

A histogram shows the frequency of fires according to their sizes: at least 50 trees burned, at least 100, at least 150, etc. This histogram should follow a power law.

Will fire breaks naturally grow? Will the percentage of patches in the have-tree state stabalize under certain settings for fertility? Will the forest naturally find the optimal tree covering?

We can add adaptation to our patches as follows: when a patch is updated it firts counts the number of burned neighbors. If this number is above some critical number, for example, if more than four neighbors are in the burned state, then the patch decreases its fertility by some adjustable fertility-adaptation-factor. Otherwise the patch increases its fertility by this factor. Be careful. The fertility of a patch should always be a number between 0 and 1. After updating its fertility, the patch either grows a tree, burns, or does nothing as before.

What setting for the fertility-adaptation-factor produces the largest number of trees?

PROJECT: AVALANCHE!
--------------
Assume the state of each patch represents the number of grains of sand located on that patch. The color of a patch should be a shade of brown that reflects its state. A darker shade indicates more sand.

In addition to its usual duties, the update-model procedure drops a single grain of sand on 10 randomly chosen patches. (There's nothing magic about 10.)

If the state of a patch exceeds num-states, it distributes one grain of sand to each patch in my-neighborhood. (Of course this may cause these patches to go over the limit.) 

Use the histogram function to create a histogram showing the numbers of patches with 0, 1, 2, ..., num-states grains of sand.

Note that the histogram indicates an exponential growth in large patches. This is an example of the power law.

RELATED MODELS
--------------
This section could give the names of models in the NetLogo Models Library or elsewhere which are of related interest.


CREDITS AND REFERENCES
----------------------
Created by Jon Pearce (http://www.cs.sjsu.edu/faculty/pearce/pearce.html)
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link direction
true
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Line -7500403 true 150 150 90 180
Line -7500403 true 150 150 210 180

@#$#@#$#@