cellular automaton

A $d$ -dimensional cellular automaton is a triple $\mathcal{A}=\langle Q,\mathcal{N},f\rangle$ where:

1.

$Q$ is a nonempty set, called alphabet or set of states.
2.

$\mathcal{N}=\{\nu_{1},\ldots,\nu_{|\mathcal{N}|}\}$ is a finite, nonempty subset of $\mathbb{Z}^{d}$ , called neighborhood index.
3.

$f:Q^{\mathcal{N}}\to Q$ is the local function.

The local function of the cellular automaton $\mathcal{A}$ induces, by synchronous application at all points, a global function $F_{\mathcal{A}}$ on $d$ -dimensional configurations $c:\mathbb{Z}^{d}\to Q$ :

$F_{\mathcal{A}}(c)(z)=f\left(c(z+\nu_{1}),\ldots,c(z+\nu_{|\mathcal{N}|})% \right)=f\left(\left.{c}\right|_{z+\mathcal{N}}\right)\;.$

(1)

Observe that the global function is continuous in the product topology.

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 $1$ but not in dimension $2$ (or higher).

The definition above can be generalized to the case of a generic group $G$ acting by translations on the space $Q^{G}$ of configurations. If

$L_{g}(x)=g\cdot x\;\;\forall x\in G\;,$

(2)

then the local function $f:Q^{\mathcal{N}}\to Q$ ( $\mathcal{N}$ being a finite nonempty subset of $G$ ) induces $F_{\mathcal{A}}:Q^{G}\to Q^{G}$ via the relation

$F_{\mathcal{A}}(c)(g)=f\left(\left.{(c\circ L_{g})}\right|_{\mathcal{N}}\right% )=f\left(\left.{c}\right|_{g\mathcal{N}}\right)\;\;\forall c:G\to Q\;.$

(3)

Observe that $L=\{L_{g}\}_{g\in G}$ is a left action, while $(c,g)\mapsto c\circ L_{g}$ is a right action. It is of course possible to only use left action by defining $F_{\mathcal{A}}(c)(g)$ as $f\left(\left.{(c\circ L_{g^{-1}})}\right|_{\mathcal{N}}\right)$ instead: since this is just a reparametrization of the family $L$ , the bulk of the theory does not change.

More to come …

Title	cellular automaton
Canonical name	CellularAutomaton
Date of creation	2013-03-22 19:21:58
Last modified on	2013-03-22 19:21:58
Owner	Ziosilvio (18733)
Last modified by	Ziosilvio (18733)
Numerical id	5
Author	Ziosilvio (18733)
Entry type	Definition
Classification	msc 37B15
Classification	msc 68Q80
Defines	neighborhood (cellular automaton)
Defines	neighborhood index (cellular automaton)
Defines	local function (cellular automaton)
Defines	global function (cellular automaton)