Definition:Stencil

Definition
A quintuple $(X,S_X,S(0),\Delta,\delta)$ defines a stencil
 * 1) its index domain $X$ is discrete $X\subseteq\Z^{|X|}$ and of finite dimension $|X|\in\N$ $\land$
 * 2) its state range $S_X$ is a well-defined set $\land$
 * 3) its initial state $S(0)$ maps from index space to state space $S(0):X\to S_X$ $\land$
 * 4) its neighbourhood delta $\Delta$ is a vector of index offsets $\Delta\in\left(\Z^{|X|}\right)^{|\Delta|}$ $\land$
 * 5) its transition combinator $\delta$ is a mapping $\delta:S_X^{|\Delta|}\to S_X$ from neighbourhood states to states.

By means of the update sweep $\forall \vec x\in X: S(n+1)(\vec x)=\delta\left(\prod_{i=1}^{|\Delta|}S(n)(\vec x+\Delta_i)\right)$ this induces a stencil evolution $\N\ni n\mapsto S(n)\in(X\to S_X)$ from the initial state $S(0)$.