Simulation in the aTAM

Formal Definition

From this point on, let \(T\) be a \(d\)-dimensional tile set, and let %, for some \(d \in \{2,3\}\). \(m\in\Z^+\). An \emph{\(m\)-block supertile} over \(T\) is a partial function \(\alpha : \Z_m^d \dashrightarrow T\), where \(\Z_m = \{0,1,\ldots,m-1\}\). Note that the dimension of the \(m\)-block is implicitly defined by the dimension of \(T\). Let \(B^T_m\) be the set of all \(m\)-block supertiles over \(T\). The \(m\)-block with no domain is said to be \(\emph{empty}\). For a general assembly \(\alpha:\Z^d \dashrightarrow T\) and \((x_0,\ldots x_{d-1})\in\Z^d\), define \(\alpha^m_{x_0,\ldots x_{d-1}}\) to be the \(m\)-block supertile defined by \(\alpha^m_{x_0,\ldots, x_{d-1}}(i_0,\ldots, i_{d-1}) = \alpha(mx_0+i_0,\ldots, mx_{d-1}+i_{d-1})\) for \(0 \leq i_0, \ldots, i_{d-1}< m\). %\paragraph{Block replacement function.} For some tile set \(S\) of dimension \(d' \geq d\), a partial function \(R: B^{S}_m \dashrightarrow T\) is said to be a \emph{valid \(m\)-block supertile representation} from \(S\) to \(T\) if for any \(\alpha,\beta \in B^{S}_m\) such that \(\alpha \sqsubseteq \beta\) and \(\alpha \in \dom R\), then \(R(\alpha) = R(\beta)\).