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surface.tex
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surface.tex
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\begin{figure}[t]
\lbl{\fbox{\ssurfacelang{}}}{
\begin{tabular}{ll}
\begin{langarray}
\sexpr & \BNFeq &
\svar
\mid \snat \mid \sint
\mid \efun{\svar}{\sexpr}
\mid \efun{\tann{\svar}{\stype}}{\sexpr}
\mid \epair{\sexpr}{\sexpr} \mid
\\ & &
\eapp{\stoptional}{\sexpr}{\sexpr}
\mid \eunopt{\stoptional}{\sexpr}
\mid \ebinopt{\stoptional}{\sexpr}{\sexpr} \mid
\\ & &
\edynb{\sbnd}{\sexpr}
\mid \estab{\sbnd}{\sexpr}
\\
\stype & \BNFeq &
\tnat \mid \tint \mid \tpair{\stype}{\stype} \mid \tfun{\stype}{\stype}
\\
\stoptional & \BNFeq &
\stype \mid \tdyn
\\
\sbnd & \BNFeq &
\obnd{\sowner}{\stype}{\sowner}
\\
\sowner & \BNFeq &
\textrm{\scountable{} set of names}
\end{langarray}
&
\begin{langarray}
\sunop & \BNFeq &
\sfst \mid \ssnd
\\
\sbinop & \BNFeq &
\ssum \mid \squotient
\\
\tenv & \BNFeq & % or \stypeenv if you're not into the whole brevity thing
\cdot \mid \fcons{\tann{\svar}{\stoptional}}{\tenv}
\\
\sownerenv & \BNFeq &
\cdot \mid \fcons{\tann{\svar}{\sowner}}{\sownerenv}
\\
\snat & \BNFeq & \naturals
\\
\sint & \BNFeq & \integers
\\
\sbset & \BNFeq &
\powerset{\sbnd}
\\
\sowners & \BNFeq &
\powerset{\sowner}
\end{langarray}
\end{tabular}
}
\lbl{\fbox{$\fwellformed{\sexpr}{\stoptional}$}}{\(\begin{array}{l@{~~}l} % well-formed expression
\fwellformed{\sexpr_0}{\stype_0} &
\mbox{ iff } \,\exists\,\sowner_0.\,\sowner_0 \sWL \sexpr_0
\mbox{ and } \sWT \sexpr_0 : \stype_0
\\
\fwellformed{\sexpr_0}{\tdyn} &
\mbox{ iff } \,\exists\,\sowner_0.\,\sowner_0 \sWL \sexpr_0
\mbox{ and } \sWT \sexpr_0 : \tdyn
\end{array}\)}{}
\medskip
\lbl{\fbox{$\sownerenv; \sowner \sWL \sexpr$} \missingrules{}}{\begin{mathpar} % well-named components (selected rules)
\inferrule*{
\tann{\svar_0}{\sowner_0} \in \sownerenv_0
}{
\sownerenv_0; \sowner_0 \sWL \svar_0
}
\inferrule*{
\fcons{\tann{\svar_0}{\sowner_0}}{\sownerenv_0}; \sowner_0 \sWL \sexpr_0
}{
\sownerenv_0; \sowner_0 \sWL \efun{\svar_0}{\sexpr_0}
}
\inferrule*{
\fcons{\tann{\svar_0}{\sowner_0}}{\sownerenv_0}; \sowner_0 \sWL \sexpr_0
}{
\sownerenv_0; \sowner_0 \sWL \efun{\tann{\svar_0}{\stype_0}}{\sexpr_0}
}
\inferrule*{
\sownerenv_0; \sowner_1 \sWL \sexpr_0
}{
\sownerenv_0; \sowner_0 \sWL \edynb{\obnd{\sowner_0}{\stype_0}{\sowner_1}}{\sexpr_0}
}
\end{mathpar}}
\medskip
\lbl{\fbox{$\tenv \sWS \sexpr : \stype$} \missingrules{}}{\begin{mathpar} % static typing (selected rules)
% \inferrule*{
% \fmember{\tann{\svar_0}{\stype_0}}{\tenv_0}
% }{
% \tenv_0 \sWS \svar_0 : \stype_0
% }
%
% \inferrule*{
% }{
% \tenv_0 \sWS \snat_0 : \tnat
% }
%
% \inferrule*{
% }{
% \tenv_0 \sWS \sint_0 : \tint
% }
\inferrule*{
\fcons{\tann{\svar_0}{\stype_0}}{\tenv_0} \sWS \sexpr_0 : \stype_1
}{
\tenv_0 \sWS \efun{\tann{\svar_0}{\stype_0}}{\sexpr_0} : \tfun{\stype_0}{\stype_1}
}
% \inferrule*{
% \tenv_0 \sWS \sexpr_0 : \stype_0
% \\
% \tenv_0 \sWS \sexpr_1 : \stype_1
% }{
% \tenv_0 \sWS \epair{\sexpr_0}{\sexpr_1} : \tpair{\stype_0}{\stype_1}
% }
%
% \inferrule*{
% \tenv_0 \sWS \sexpr_0 : \tfun{\stype_0}{\stype_1}
% \\
% \tenv_0 \sWS \sexpr_1 : \stype_0
% }{
% \tenv_0 \sWS \eapp{\stype_1}{\sexpr_0}{\sexpr_1} : \stype_1
% }
%
% \inferrule*{
% \tenv_0 \sWS \sexpr_0 : \stype_0
% \\
% \sDelta(\sunop, \stype_0) \feq \stype_1
% }{
% \tenv_0 \sWS \eunop{\sexpr_0} : \stype_1
% }
%
% \inferrule*{
% \tenv_0 \sWS \sexpr_0 : \stype_0
% \\
% \tenv_0 \sWS \sexpr_1 : \stype_1
% \\
% \sDelta(\sbinop, \stype_0, \stype_1) \feq \stype_2
% }{
% \tenv_0 \sWS \ebinop{\sexpr_0}{\sexpr_1} : \stype_2
% }
\inferrule*{
\tenv_0 \sWS \sexpr_0 : \stype_0
\\
\stype_0 \subteq \stype_1
}{
\tenv_0 \sWS \sexpr_0 : \stype_1
}
\inferrule*{
\tenv_0 \sWD \sexpr_0 : \tdyn
}{
\tenv_0 \sWS \edyn{\obnd{\sowner_0}{\stype_0}{\sowner_1}}{\sexpr_0} : \stype_0
}
\end{mathpar}}
\medskip
\lbl{\fbox{$\tenv \sWD \sexpr : \tdyn$} \missingrules{}}{\begin{mathpar} % dynamic typing (selected rules)
% \inferrule*{
% \fmember{\svar_0}{\tenv_0}
% }{
% \tenv_0 \sWD \svar_0 : \tdyn
% }
%
% \inferrule*{
% }{
% \tenv_0 \sWD \sint_0 : \tdyn
% }
\inferrule*{
\fcons{\tann{\svar_0}{\tdyn}}{\tenv_0} \sWD \sexpr_0 : \tdyn
}{
\tenv_0 \sWD \efun{\svar_0}{\sexpr_0} : \tdyn
}
% \inferrule*{
% \tenv_0 \sWD \sexpr_0 : \tdyn
% \\
% \tenv_0 \sWD \sexpr_1 : \tdyn
% }{
% \tenv_0 \sWD \epair{\sexpr_0}{\sexpr_1} : \tdyn
% }
\inferrule*{
\tenv_0 \sWD \sexpr_0 : \tdyn
\\
\tenv_0 \sWD \sexpr_1 : \tdyn
}{
\tenv_0 \sWD \eapp{\tdyn}{\sexpr_0}{\sexpr_1} : \tdyn
}
% \inferrule*{
% \tenv_0 \sWD \sexpr_0 : \tdyn
% }{
% \tenv_0 \sWD \eunop{\sexpr_0} : \tdyn
% }
%
% \inferrule*{
% \tenv_0 \sWD \sexpr_0 : \tdyn
% \\
% \tenv_0 \sWD \sexpr_1 : \tdyn
% }{
% \tenv_0 \sWD \ebinop{\sexpr_0}{\sexpr_1} : \tdyn
% }
\inferrule*{
\tenv_0 \sWS \sexpr_0 : \stype_0
}{
\tenv_0 \sWD \esta{\obnd{\sowner_0}{\stype_0}{\sowner_1}}{\sexpr_0} : \tdyn
}
\end{mathpar}}
\medskip
\lbl{\fbox{$\stype \subteq \stype$}}{\vspace{-4mm}\begin{mathpar}
\inferrule*{
}{
\tnat \subteq \tint
}
\inferrule*{
\stype_0 \subteq \stype_2
\\
\stype_1 \subteq \stype_3
}{
\tpair{\stype_0}{\stype_1} \subteq \tpair{\stype_2}{\stype_3}
}
\inferrule*{
\stype_2 \subteq \stype_0
\\
\stype_1 \subteq \stype_3
}{
\tfun{\stype_0}{\stype_1} \subteq \tfun{\stype_2}{\stype_3}
}
\inferrule*{
}{
\stype_0 \subteq \stype_0
}
\end{mathpar}}
\Description[Grammar and typing judgments.]{Defines the expressions for
a model language, a labeling judgment, two typing judgments, and
a subtyping relation.
}
\caption{\ssurfacelang{}}
\label{fig:surface-language}
\vspace{-3mm}
\end{figure}
\noindent Our model builds on the previous work of
\citeN{gf-icfp-2018} who, in turn, employ
\citeN{mf-toplas-2009}'s multi-language framework for modeling the syntax,
types, and semantics of a mixed-typed language.
In particular, we interpret one ``surface'' syntax and mixed
type system (\figref{fig:surface-language}) in three different ways. This
section presents the common basis.
The set of surface expressions~$\sexpr$ includes numbers, tuples, and anonymous
functions, illustrative of the atomic values, data structures, and higher-order values.
Expressions may be combined
through function application ($\sapp$), operator application ($\sunop$ and
$\sbinop$), and boundary terms ($\sdyn$ and $\ssta$).
Boundary terms divide a program into named components.
Each component is either statically or dynamically typed.
A boundary term combines a boundary specification ($\sbnd$, for instance
$\obnd{\sowner_0}{\stype_0}{\sowner_1}$) and a sender expression.
For example, the boundary term $(\edynb{\obnd{\sowner_0}{\tnat}{\sowner_1}}{(\efun{\svar_0}{\svar_0})})$
embeds a dynamically-typed sender component named $\sowner_1$,
consisting of the function $(\efun{\svar_0}{\svar_0})$,
into a statically-typed context named $\sowner_0$.
While $\tnat$ seems to disagree with the enclosed function value,
the term is grammatically correct and well-typed.
Every function application and operator application comes with the type
$\stoptional$ of its expected result.\footnote{These annotations serve two
purposes: one essential, one convenient. On elimination forms ($\sapp$,
$\sfst$, and $\ssnd$) they are essential for the first-order
interpretations, because those must check the output of certain
elimination forms in typed components and thus enforce the type required
by the context. Binary operations have annotations to conveniently
disambiguate statically-typed and dynamically-typed redexes.
All these annotations could
be inferred from the type derivation of an unannotated surface program.} In a
statically-typed component, any type $\stype$ may be acceptable. Within a
dynamically-typed component, the mark $\tdyn$ of a un(i)typed value is the
only appropriate declaration.
Note that $\tdyn$ is not a dynamic type~\cite{t-popl-1990}.
Our model is thus directly representative of languages
such as Dart~2 ({\small\shorturl{https://}{dart.dev/dart-2}}) and Typed Racket,
and of the cast calculi employed by many true gradual
languages~\cite{svcb-snapl-2015,cs-popl-2017}.
An expression is well-formed if it uses component names correctly and
is well-typed. Figure~\ref{fig:surface-language} defines the relevant
judgments. The first judgment, $\sWL$, checks that components are
named consistently and that the variables defined in one component are never
directly accessed by another component.
This judgment applies to both static and
dynamic code.
Type-correctness uses two judgments: one for static typing
($\stypeenv \sWS \sexpr : \stype$) and one for dynamic typing ($\stypeenv
\sWD \sexpr : \tdyn$).
These judgments depend on one another to handle
boundary terms.
For example, a $\ssta$ boundary has type $\tdyn$ if
its subexpression has the type in the boundary specification. The static typing
judgment depends on a notion of subtyping ($\subteq$)---based on a subset
relationship between the $\tnat$ and $\tint$ types---because we firmly
embrace the idea that a type system for untyped code must
have true union types~\cite{tf-icfp-2010,cl-icfp-2017,tfffgksst-snapl-2017}.
Subtyping is incorporated to the typing
judgment via a subsumption rule to keep the presentation brief.