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Dmitry Boulytchev 2021-02-10 00:56:28 +03:00
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\begin{table}[t]
\begin{tabular}{cccl}
notation & instances & definition & comments \\[1mm]
\hline\\
$\inbr{\bullet,\,\bullet,\,\dots}$ & & & pairs and tuples\\[1mm]
$X^\bot$ & & & $X\cup\{\bot\}$\\[1mm]
$X^*$ & & & lists of elements of $X$\\[1mm]
$\mathscr X$ & $x,\,y,\,z,\,\dots$ & & variables \\[1mm]
$\mathscr E$ & & & expressions \\[1mm]
$\mathscr T$ & $\con{C},\,\con{D},\,\dots$ & & tags (constructors) \\[1mm]
$\Sigma$ & $\sigma$ & $\mathscr X\to\mathscr V$ & bindings (a partial map from\\
& & & variables to values) \\[1mm]
$\Sigma_{\mathscr X}$ & $\inbr{S,\,\sigma}$ & $2^{\mathscr X}\times\Sigma$ & local scopes (sets of variables\\
& & & and their bindings) \\[1mm]
$St$ & $\inbr{\sigma_g,\,ss}$ & $\Sigma\times\Sigma^*_{\mathscr X}$ & states (global bindings and stacks\\
& & & of local scopes) \\[1mm]
$\mathcal W$ & $w$ & & worlds\\[1mm]
$\mathcal L$ & $l$ & & locations \\[1mm]
$\mathcal M$ & $\mu$ & $\mathcal L\to\mathcal C$ & abstract memory (a partial map from\\
& & & locations to composite values) \\[1mm]
$\mathcal C$ & $c$ & $St\times\mathcal{M}\times\mathcal{W}$ & configurations (state, memory and world)\\[2mm]
$\mathcal P$ & & $\mathbb Z^\bot\uplus$ & primitive values (integer or default\\
& & $\primi{ref}{\mathscr{X}}\uplus$ & value, references to variables or\\
& & $\primi{elemRef}{\mathcal L\;\mathbb N}$ & array elements)\\[2mm]
$\mathcal F$ & & $\mathscr X^*\times\mathscr E \times \mathbb N$ & function values (argument names, body,\\
& & & nesting level)\\[1mm]
$\mathcal V$ & $v$ & $\mathcal P\uplus \mathcal L$ & values (primitive values or locations) \\[1mm]
$\mathcal C$ & & $Arr\uplus Sexp \uplus Clo$ & composite values (arrays, S-expressions, \\
& & & or closures) \\[1mm]
$Arr$ & & $\mathbb N\times (\mathbb N\to\mathcal V)$ & arrays (length and element function) \\[1mm]
$Sexp$ & & $\mathscr T \times Arr$ & S-expressions (tag and array of subvalues) \\[1mm]
$Clo$ & & $\mathscr X^* \times\mathscr E\times\Sigma^*_{\mathscr X}$ & closures (argument names, function body\\
& & & and a stack of local scopes)
\end{tabular}
\caption{Basic Categories}
\label{categories}
\end{table}