CellularAutomaton

CellularAutomaton[rule,init,t]

generates a list representing the evolution of the cellular automaton with the specified rule from initial condition init for t steps.

CellularAutomaton[rule,init]

gives the result of evolving init for one step.

CellularAutomaton[rule,init,{tspec,xspec,}]

gives only those parts of the evolution specified by tspec, xspec, etc.

CellularAutomaton[rule,init,{t,All,}]

includes at each step all cells that could be affected over the course of t steps.

CellularAutomaton[rule]

is an operator form of CellularAutomaton that represents one step of evolution.

Details

• Possible forms for rule are:
•  n , , elementary rule with rule number n {n,k} general nearest‐neighbor rule with k colors {n,k,r} general rule with k colors and range r {n,k,{r1,r2,…,rd}} d‐dimensional rule with neighborhood {n,k,{{off1},{off2},…,{offs}}} rule with neighbors at specified offsets {n,k,rspec,s} order‐s rule {n,{k,1}} k‐color nearest‐neighbor totalistic rule {n,{k,1},r} k‐color range-r totalistic rule {n,{k,{wt1,wt2,…}},rspec} rule in which neighbor i is assigned weight wti {lhs1->rhs1,lhs2->rhs2,…} explicit replacements for lists of neighbors {fun,{},rspec} general function fun to apply to each list of neighbors bfun Boolean function to apply to collections of neighbors CellularAutomaton[rule] operator form of a rule <|"key1"val1,"key2"val2,…|> rule specification by an association "name" named rule
• CellularAutomaton[{n,k},] is equivalent to CellularAutomaton[{n,{k,{k^2,k,1}}},].
• The following keys can be used to specify a rule given as an association:
•  "RuleNumber" n rule number "TotalisticCode" n totalistic code "OuterTotalisticCode" n outer totalistic code "GrowthCases" {g1,g2,…} make a cell 1 when gi of its neighbors are 1 "GrowthSurvivalCases" {{g1,…},{s1,…}} 1 for gi neighbors; unchanged for si "GrowthDecayCases" {{g1,…},{d1,…}} 1 for gi neighbors; 0 for di "Dimension" d overall dimension "Colors" k number of colors "Range" r range of rule "Neighborhood" type neighborhood type
• With "GrowthCases"->{g1,g2,}, a cell goes from value 0 to value 1 if it has gi neighbors that are 1; otherwise it stays the same as before.
• With "GrowthSurvivalCases"->{{g1,},{s1,}}, a cell goes from value 0 to value 1 if it has gi neighbors that are 1, maintains value 1 if it has si neighbors that are 1, and otherwise gets value 0.
• With "GrowthDecayCases"->{{g1,},{d1,}}, a cell goes from value 0 to value 1 if it has gi neighbors that are 1, gets value 0 if it has di neighbors that are 1, and otherwise stays the same.
• Possible settings for "Neighborhood" in 2D include:
•  5 or "VonNeumann" CrossMatrix[1] 9 or "Moore" BoxMatrix[1]
• For dimension d, "Neighborhood" supports "VonNeumann" and "Moore", as well as the integers and .
• Possible named cellular automaton rules given as CellularAutomaton["name",] include:
•  "Rule30" 30 "Rule90" 90 "Rule110" 110 "Code1599" {1599,{3,1}} "GameOfLife" {224,{2,{{2,2,2},{2,1,2},{2,2,2}}},{1,1}}
• Common explicit forms for 2D cellular automaton rules include:
•  {n,{k,1},{1,1}} 9‐neighbor totalistic rule {n,{k,{{0,1,0},{1,1,1},{0,1,0}}},{1,1}} 5‐neighbor totalistic rule {n,{k,{{0,k,0},{k,1,k},{0,k,0}}},{1,1}} 5‐neighbor outer totalistic rule
• The number of possible cellular automaton rules is as follows:
•  elementary rules 256 1D general rules 1D totalistic rules 2D general rules 2D 9‐neighbor totalistic rules 2D 5‐neighbor totalistic rules 2D 5‐neighbor outer totalistic rules
• Normally, all elements in init and the evolution list are integers between 0 and .
• When a general function or a replacement list is used, the elements of init and the evolution list can be any expressions. »
• Explicit replacement rules lhsi->rhsi can contain patterns.
• In a 1D cellular automaton, replacement rules or an explicit function fun are always taken to apply to a 1D list of neighbors. If the neighbors are specified by explicit offsets, they are given in the order of the offsets.
• When the neighborhood in a multidimensional cellular automaton is defined by a range specification such as {r1,r2,}, the list of neighbors is taken to be a full array with dimensions 2{r1,r2,}+1.
• If the neighbors in a multidimensional cellular automaton are specified by an explicit list of offsets, the neighbors are supplied in a one-dimensional list in the order of the offsets.
• If an explicit function fun is given, the first argument supplied to it is the list of neighbors. The second argument is the step number starting at 0.
• A complete rule specification is considered to be a pure Boolean function bfun if BooleanVariables[bfun] yields an integer v. In this case, bfun is applied to neighborhoods of v cells at each step. The neighborhoods extend Ceiling[v/2] cells to the left.
• In an orders cellular automaton, specified by {rule,kspec,rspec,s}, each step depends on s preceding steps.
• Initial conditions are constructed from init as follows:
•  {a1,a2,…} explicit list of values ai, assumed cyclic {{a1,a2,…},b} values ai superimposed on a b background {{a1,a2,…},{b1,b2,…}} values ai superimposed on a background of repetitions of b1, b2, … {{{{a11,a12,…},off1}, {{a21,…},off2},…},bspec} values aij at offsets offi on a background {{a11,a12,…},{a21,…},…} explicit list of values in two dimensions {aspec,bspec} values in d dimensions with d‐dimensional padding
• The first element of aspec is superimposed on the background at the first position in the positive direction in each coordinate relative to the origin. This means that bspec[[1,1,]] is aligned with aspec[[1,1,]].
• CellularAutomaton[rule,init,t] generates an evolution list of length .
• For an orders cellular automaton, init is a list giving the initial s steps in the evolution of the system.
• Time specifications tspec in {tspec,xspec,} can be as follows:
•  t all steps 0 through t {t} a list containing only step t {{t}} step t alone {t1,t2} steps t1 through t2 {t1,t2,dt} steps t1, t1+dt, …
• The initial condition is considered to be at step 0.
• CellularAutomaton[rule,init,{tspec}] uses the default Automatic for xspec.
• Space specifications xspec can be as follows:
•  All all cells that can be affected by the specified initial condition Automatic all cells in the region that differs from the background 0 cell aligned with beginning of aspec x cells at offsets up to x on the right -x cells at offsets up to x on the left {x} cell at offset x to the right {-x} cell at offset x to the left {x1,x2} cells at offsets x1 through x2 {x1,x2,dx} cells x1, x1+dx, …
• In one dimension, the first element of aspec is taken by default to have space offset 0.
• In any number of dimensions, aspec[[1,1,1,]] is taken by default to have space offset {0,0,0,}.
• Each element of the evolution list produced by CellularAutomaton is always the same size.
• With an initial condition specified by an aspec of width , the region that can be affected after steps by a cellular automaton with a rule of range has width .
• If no bspec background is specified, space offsets of All and Automatic will include every cell in aspec.
• A space offset of All includes all cells that can be affected by the initial condition.
• A space offset of Automatic can be used to trim off background from the sides of a cellular automaton pattern.
• In working out how wide a region to keep, Automatic only looks at results on steps specified by offt.
• CellularAutomaton[rule][init] is equivalent to CellularAutomaton[rule,init].

Examples

open allclose all

Basic Examples(3)

Run rule 30 for 2 steps:

 In[1]:=
 Out[1]=

Run for 50 steps from a single 1 on a background of 0s:

 In[1]:=
 Out[1]=

Generate an icon for a cellular automaton rule:

 In[1]:=
 Out[1]=