Definition:Stencil

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Definition

A quintuple $\tuple {X, S_X, \map S 0, \Delta, \delta}$ defines a stencil if and only if:

\((1)\)	$:$		Index Domain:					$X$ is discrete: $X \subseteq \Z^{\size X}$ and of finite dimension: $\size X \in \N$
\((2)\)	$:$		State Range:					$S_X$ is a well-defined set
\((3)\)	$:$		Initial State:					$\map S 0$ maps from index space to state space $\map S 0: X \to S_X$
\((4)\)	$:$		Neighbourhood Delta:					$\Delta$ is a vector of index offsets: $\Delta \in \paren {\Z^{\size X} }^{\size \Delta}$
\((5)\)	$:$		Transition Combinator:					$\delta$ is a mapping $\delta: S_X^{\size \Delta} \to S_X$ from neighbourhood states to states

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