Cellular Automata/Mathematical Model

Formally, a cellular automaton is represented by the 4-tuple where:

  • is the finite or infinite lattice
  • is a finite set of cell states or values
  • is the finite neighborhood
  • is the local transition function defined by the transition table or the rule

The lattice is a finite or infinite discrete regular grid of cells on a finite number of dimensions. Each cell is defined by its discrete position (an integer number for each dimension) and by its discrete value (one of a finite set of integers). Time is also discrete. The future state of a cell (time ) is a function of the present state (time ) of a finite number of cells surrounding the observed cell called the neighborhood.

One dimensional first order cellular automataEdit

For the sake of readability the next definitions focus on one dimensional first order cellular automata.

Lattice, cell and configurationEdit

The infinite global state is a configuration  .   is the finite set   of cell states  , for formalization purposes the states are enumerated  . The lattice   is the infinite cyclic group of integers  . The position of each cell inside the lattice is described by the position index  . Configurations are usually written as strings.

 

The finite global state is a finite configuration  , where   is a finite lattice, a finite set   of   integers.

 

Finite configurations and their parts can be generally written as strings denominated by small Greek letters from the beginning of the alphabet ( ,  , ...).

The number notation for stringsEdit

Strings can be compactly written as numbers. A string of   characters   from a set of   symbols is translated into a  -digit base   number. Usually strings are indexed from the left to the right but for the number notation indexing from right to left is more intuitive.

 

Neighborhood, local transition function and ruleEdit

 
The neighborhood size and position

NeighborhoodEdit

The neighborhood   of size   is defined by the set of relative positions inside the configuration.

 

By applying the set   onto an observed cell   the neighborhood of this cell is obtained.

 

The name neighborhood can be used for booth the set of relative distances and for the actual substring of cells related to an observed cell.

A compact representation of the neighborhood value   is a single integer defined as a number of   digits base  .

 
 
The neighborhood cell indexing and the local transition function

For definitions of common neighborhoods see Neighborhoods.

Local transition functionEdit

The local transition function

 

calculates the value of a single future cell   from the neighborhood of the observed cell in the present.

 

The transition table defines the local transition function by listing the output value for each input value.

 n  -> f(n)
-----------
000 -> 0
001 -> 0
........
111 -> 0

The rule   is a compact representation of the local transition function. It is a single integer defined as a number of   digits base  .

 
See also

Global transition functionEdit

The global dynamics of CA are described by the global transition function

 

  translates the current (present) configuration   into the next (future) configuration  

 

The global transition function   is defined by the local transition function   as

 

Finite lattices and lattice boundariesEdit

 
The lattice boundary sizes (left and right)

Infinite cellular automata have no boundary, so its boundary description   is omitted. But there is no way to simulate an infinite system using a finite system. The simulation must focus on a finite part of length  .

The neighborhood used in the local transition function oversteps the lattice boundary for   cells at the left and   cells at the right.

There are two common solutions to the overstepping problem:

  1. the lattice is wrapped into a circle (torus for 2D CA)
  2. the values of the overstepping parts of the neighborhood are defined explicitly as the boundary  

Cyclic boundariesEdit

Cyclic boundaries are frequently used as there is no need to explicitly define the boundary value and no external information is introduced into the CA that could otherwise cause interference at the boundaries.

The state of a finite lattice cellular automata is a configuration in the lattice  , where   is a cyclic group of integers modulo   ( ).

 

The cyclic position index is calculated as

 

Explicitly defined boundariesEdit

 
The lattice boundary (left and right)

Explicitly defined boundaries are less common as the simple constant values are useful only for CA where we observe events on a quiescent background with period 1. The boundary can be defined as a single set (the left and the right part combined) of cell values of length   (there is no boundary cell with index 0)

 

For space and time periodic quiescent backgrounds time dependent boundaries can be used  .

GeneralizationsEdit

Multidimensional cellular automataEdit

 
Neighborhood in a 2D lattice

The definition of n-dimensional CA is similar to that of one dimensional CA, the lattice becomes n-dimensional and   and   become vectors of length  .

2D cellular automataEdit

The 2D lattice can be tiled with cells in different ways:

2D cellular automata are often used to simulate real dynamic systems (fluid and gas dynamics)

See also

Cellular automata on groupsEdit

One further generalization of the concept of a CA extends the n-dimensional construct. Given a finitely generated group,  , and a alphabet,  , we may define the configuration space to be  . That is, each configuration is a map from   into  . If   is abelian, then the group is isomorphic to some quotient space of   and may be regarded as a n-dimensional lattice with possibly periodic boundary conditions. This space admits a group action, where   where   is the inverse of  . Any finitely generated group is a metric space, in which the distance between any two elements,   can be defined to be the minimum length of the set of paths connecting   and   on the groups Cayley graph. We define a metric on the configuration space,   to be 0 if the two configurations are identical, and the infimum of   over the set of   such that   and   disagree at   and where   denotes the identity element of the group. We define a cellular automata to be a continuous mapping,  , that commutes with the group action and an initial configuration  . The evolution of the system is defined by  .

Higher order cellular automataEdit

The CA is of higher order if not only the present but past configurations too are used to calculate the future. A second-order local transition function is defined as

 

Second-order local transition functions are often used to construct reversible rules.