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app3.tex
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%\appendix
\section{Appendix: The Initial Static Basis}
\label{init-stat-bas-app}
We\index{73.1} shall indicate components of the initial basis by the subscript 0.
The initial static basis is
\[ \B_0\ =\ (\M_0,\T_0),\F_0,\G_0,\E_0\]
where
\begin{itemize}
\item $\M_0\ =\ \emptyset$
\item $\T_0\ =\ \{\BOOL,\INT,\REAL,\STRING,\LIST,\REF,\EXCN,\INSTREAM,\OUTSTREAM\}$
\item $\F_0\ =\ \emptymap$
\item $\G_0\ =\ \emptymap$
\item $\E_0\ =\ \longE{0}$
\end{itemize}
The members of $\T_0$ are type names, not type constructors; for convenience
we have used type-constructor identifiers
to stand also for the type names which are bound to them in the initial
static type environment $\TE_0$. Of these type names,
~\LIST~ and ~\REF~
have arity 1, the rest have arity 0; all except \EXCN, \INSTREAM~
and ~\OUTSTREAM~ admit equality.
The components of $\E_0$ are as follows:
\begin{itemize}
\item $\SE_0\ =\ \emptymap$
\item $\VE_0$ is shown in Figures~\ref{stat-ve} and \ref{stat-veio}. Note that
$\Dom\VE_0$ contains those identifiers ({\tt true},{\tt false},{\tt nil},
\verb+::+) which are basic value constructors,
for reasons discussed in Section~\ref{stat-proj}.
$\VE_0$ also includes $\EE_0$, for the same reasons.
\item $\TE_0$ is shown in Figure~\ref{stat-te}. Note that the type
structures in $\TE_0$ contain the type schemes of all basic value
constructors.
\item $\Dom\EE_0\ =\ \BasExc$~, the set of basic exception names listed in
Section~\ref{bas-exc}.
In each case the associated type is ~\EXCN~, except that
~$\EE_0({\tt Io})=\STRING\rightarrow\EXCN$.
\end{itemize}
\begin{figure}
\begin{tabular}{|rl|rl|}
\multicolumn{2}{c}{NONFIX}& \multicolumn{2}{c}{INFIX}\\
\hline
$\var$ & $\mapsto\ \tych$
& $\var$ & $\mapsto\ \tych$\\
\hline
{\tt map} & $\mapsto\ \forall\atyvar\ \btyvar.\ (\atyvar\to\btyvar)\to$
& \multicolumn{2}{l|}{Precedence 7 :} \\
& \qquad$\atyvar\ \LIST\to\btyvar\ \LIST$
& \verb+/+ & $\mapsto\ \REAL\ \ast\ \REAL
\to\REAL$\\
{\tt rev} & $\mapsto\ \forall\atyvar.\ \atyvar\ \LIST\to\atyvar\ \LIST$
& {\tt div} & $\mapsto\ \INT\ \ast\ \INT\to\INT$\\
{\tt not} & $\mapsto\ \BOOL\to\BOOL$
& {\tt mod} & $\mapsto\ \INT\ \ast\ \INT\to\INT$\\
\verb+~+ & $\mapsto\ \NUM\to\NUM$
& \verb+*+ & $\mapsto\ \NUM\ \ast\ \NUM\to\NUM$\\
{\tt abs} & $\mapsto\ \NUM\to\NUM$
& \multicolumn{2}{l|}{Precedence 6 :} \\
{\tt floor}& $\mapsto\ \REAL\to\INT$
& \verb-+- & $\mapsto\ \NUM\ \ast\ \NUM\to\NUM$\\
{\tt real} & $\mapsto\ \INT\to\REAL$
& \verb+-+ & $\mapsto\ \NUM\ \ast\ \NUM\to\NUM$\\
{\tt sqrt} & $\mapsto\ \REAL\to\REAL$
& \verb+^+ & $\mapsto\ \STRING\ \ast\ \STRING
\to\STRING$\\
{\tt sin} & $\mapsto\ \REAL\to\REAL$
& \multicolumn{2}{l|}{Precedence 5 :} \\
{\tt cos} & $\mapsto\ \REAL\to\REAL$
& \verb+::+ & $\mapsto\ \forall\atyvar.
\atyvar\;{\ast}\;\atyvar\;\LIST
\to\atyvar\;\LIST$\\
{\tt arctan}
& $\mapsto\ \REAL\to\REAL$
& \verb+@+ & $\mapsto\ \forall\atyvar.\
\atyvar\ \LIST\ $\\
{\tt exp} & $\mapsto\ \REAL\to\REAL$
& & $\qquad\ast\ \atyvar\ \LIST\to
\atyvar\ \LIST$\\
{\tt ln} & $\mapsto\ \REAL\to\REAL$
& \multicolumn{2}{l|}{Precedence 4 :}\\
{\tt size} & $\mapsto\ \STRING\to\INT$
& \verb+=+ & $\mapsto\ \forall\aetyvar.\
\aetyvar\ \ast\ \aetyvar\to\BOOL$\\
{\tt chr} & $\mapsto\ \INT\to\STRING$
& \verb+<>+ & $\mapsto\ \forall\aetyvar.\
\aetyvar\ \ast\ \aetyvar\to\BOOL$\\
{\tt ord} & $\mapsto\ \STRING\to\INT$
& \verb+<+ & $\mapsto\ \NUM\ \ast\ \NUM
\to\BOOL$\\
{\tt explode}
& $\mapsto\ \STRING\to\STRING\ \LIST$
& \verb+>+ & $\mapsto\ \NUM\ \ast\ \NUM
\to\BOOL$\\
{\tt implode}
& $\mapsto\ \STRING\ \LIST\to\STRING$
& \verb+<=+ & $\mapsto\ \NUM\ \ast\ \NUM
\to\BOOL$\\
\verb+!+ & $\mapsto\ \forall\atyvar.\ \atyvar\ \REF\to\atyvar$
& \verb+>=+ & $\mapsto\ \NUM\ \ast\ \NUM
\to\BOOL$ \\
\REF & $\mapsto\ \forall\ \aityvar\ .\ \aityvar\to\aityvar\ \REF$
& \multicolumn{2}{l|}{Precedence 3 :} \\
{\tt true} & $\mapsto\ \BOOL$
& \verb+:=+ & $\mapsto\ \forall\atyvar.\
\atyvar\ \REF\ \ast\ \atyvar\to\UNIT$\\
{\tt false}& $\mapsto\ \BOOL$
& {\tt o} & $\mapsto\ \forall\atyvar\ \btyvar\
\ctyvar.\ (\btyvar\to\ctyvar)$\\
{\tt nil} & $\mapsto\ \forall\atyvar.\ \atyvar\ \LIST$
& & \qquad
$\ast\ (\atyvar\to\btyvar)\to(\atyvar\to\ctyvar) $ \\
\hline
\end{tabular}
\vspace{3pt}
Notes:
\begin{itemize}
\item In type schemes we have taken the liberty of writing
$\ty_1\ast\ty_2$ in place of
$\{\mbox{\tt 1}\mapsto\ty_1,\mbox{\tt 2}\mapsto\ty_2\}$.
\item An identifier with type involving ~\NUM~ stands for two functions --
one in which ~\NUM~ is replaced by ~\INT~ in its type,
and another in which ~\NUM~ is replaced by ~\REAL~ in its type.
Sometimes an explicit type constraint will be needed if the
surrounding text does not determine the type of a particular
occurrence of \verb-+- (for example). For this purpose, the
surrounding text is no larger than the enclosing top-level
declaration; an implementation may require that a smaller
context determines the type.
%In the case
%that both types can be inferred for an occurrence of the identifier, an
%explicit type constraint is needed to determine which type is intended.
%version 2: \item The type schemes associated with special
%constants are given in Figure~\ref{stat-te} which shows the initial
%static type environment.
\end{itemize}
\caption{Static $\VE_0$ (except for Input/Output and $\EE_0$)\index{74}}
\label{stat-ve}
\end{figure}
\begin{figure}
\begin{center}
\begin{tabular}{|rl|}
\hline
$\var$ & $\mapsto\ \tych$\\
\hline
{\tt std\_in} & $\mapsto\ \INSTREAM$\\
{\tt open\_in} & $\mapsto\ \STRING\to\INSTREAM$\\
{\tt input} & $\mapsto\ \INSTREAM\ \ast\ \INT\to\STRING$\\
{\tt lookahead} & $\mapsto\ \INSTREAM\to\STRING$\\
{\tt close\_in} & $\mapsto\ \INSTREAM\to\UNIT$\\
{\tt end\_of\_stream}
& $\mapsto\ \INSTREAM\to\BOOL$\\
\multicolumn{2}{|c|}{}\\
{\tt std\_out} & $\mapsto\ \OUTSTREAM$\\
{\tt open\_out} & $\mapsto\ \STRING\to\OUTSTREAM$\\
{\tt output} & $\mapsto\ \OUTSTREAM\ \ast\ \STRING\to\UNIT$\\
{\tt close\_out} & $\mapsto\ \OUTSTREAM\to\UNIT$\\
\hline
\end{tabular}
\end{center}
\vspace{3pt}
\caption{Static $\VE_0$ (Input/Output)\index{75.1}}
\label{stat-veio}
\end{figure}
\begin{figure}
\begin{center}
\begin{tabular}{|rll|}
\hline
$\tycon$ & $\mapsto\ \{\ \typefcn$, & $\{\con_1\mapsto\tych_1,\ldots,\con_n\mapsto\tych_n\}\ \}\quad (n\geq0)$\\
\hline
\UNIT & $\mapsto\ \{\ \Lambda().\{ \}$,
& $\emptymap\ \}$ \\
\BOOL & $\mapsto\ \{\ \BOOL$, & $\{\TRUE\mapsto\BOOL,
\ \FALSE\mapsto\BOOL\}\ \}$\\
\INT & $\mapsto\ \{\ \INT$, & $\{\}\ \}$\\
\REAL & $\mapsto\ \{\ \REAL$, & $\{\}\ \}$\\
\STRING & $\mapsto\ \{\ \STRING$, & $\{\}\ \}$\\
%version 2: \INT & $\mapsto\ \{\ \INT$, & $\{i\mapsto\INT\ ;\
% i$ an integer constant$\}\ \}$\\
%\REAL & $\mapsto\ \{\ \REAL$, & $\{r\mapsto\REAL\ ;\
% r$ a real constant$\}\ \}$\\
%\STRING & $\mapsto\ \{\ \STRING$, & $\{s\mapsto\STRING\ ;\
% s$ a string constant$\}\ \}$\\
\LIST & $\mapsto\ \{\ \LIST$, & $\{\NIL\mapsto\forall\atyvar\ .\ \atyvar\ \LIST$,\\
& & $\ \mbox{\verb+::+}\ \mapsto\forall\atyvar\ .
\ \atyvar\ast\atyvar\ \LIST
\to\atyvar\ \LIST\}\ \}$\\
%\LIST & \multicolumn{2}{l|}{$\mapsto\ \{\ \LIST, \{\NIL\mapsto
% \forall\alpha.\alpha\LIST,\
% \ml{::}\mapsto\forall\alpha.
% \alpha\ast\alpha\LIST
% \to\alpha\LIST\}\ \}$
% }\\
\REF & $\mapsto\ \{\ \REF$, & $\{\REF\mapsto\forall\ \aityvar\ .\
\aityvar\to\aityvar\ \REF\}\ \}$\\
\EXCN & $\mapsto\ \{\ \EXCN$, & $\emptymap\ \}$\\
\INSTREAM & $\mapsto\ \{\ \INSTREAM$,& $\emptymap\ \}$ \\
\OUTSTREAM & $\mapsto\ \{\ \OUTSTREAM$,& $\emptymap\ \}$ \\
\hline
\end{tabular}
\end{center}
\caption{Static $\TE_0$\index{75.2}}
\label{stat-te}
\end{figure}