A -dimensional cellular automaton is a triple where:
The local function of the cellular automaton induces, by synchronous application at all points, a global function on -dimensional configurations :
Cellular automata are a Turing-complete model of computation. In fact, given a Turing machine, it is straightforward to construct a cellular automaton that emulates it in real time. On the other hand, given a one-dimensional cellular automaton, there exists a Turing machine that emulates it in linear time with respect to the size of the input.
Cellular automata make good tools for qualitative analysis. In fact, given the description of a cellular automaton, it is straightforward to write a computer program that implements its features. Also, several phenomena in different fields of sciences — from physics to biology to sociology — can be described in terms of finite-range interactions between discrete agents.
On the other hand, retrieving the properties of the global dynamics from the local description is usually a very difficult task, and may depend on the dimension. As an example, reversibility of the global function is decidable in dimension but not in dimension (or higher).
then the local function ( being a finite nonempty subset of ) induces via the relation
Observe that is a left action, while is a right action. It is of course possible to only use left action by defining as instead: since this is just a reparametrization of the family , the bulk of the theory does not change.
More to come …
|Date of creation||2013-03-22 19:21:58|
|Last modified on||2013-03-22 19:21:58|
|Last modified by||Ziosilvio (18733)|
|Defines||neighborhood (cellular automaton)|
|Defines||neighborhood index (cellular automaton)|
|Defines||local function (cellular automaton)|
|Defines||global function (cellular automaton)|