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Typing

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Daniel Hillerström
2019-12-20 17:01:35 +00:00
parent f7ab2dbea6
commit aaa862b1c5
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8
macros.tex
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@@ -66,6 +66,11 @@
\newcommand{\False}{\mathsf{false}} \newcommand{\False}{\mathsf{false}}
\newcommand{\eff}{!} \newcommand{\eff}{!}
\newcommand{\typ}[2]{#1 \vdash #2}
\newcommand{\typv}[2]{#1 \vdash #2}
\newcommand{\typc}[3]{#1 \vdash #2 \eff #3}
\newcommand{\FTV}{\ensuremath{\mathrm{FTV}}}
%% Handler projections. %% Handler projections.
\newcommand{\hret}{H^{\mathrm{val}}} \newcommand{\hret}{H^{\mathrm{val}}}
@@ -82,11 +87,12 @@
%% Labels %% Labels
%% %%
\newcommand{\slab}[1]{\textrm{#1}} \newcommand{\slab}[1]{\textrm{#1}}
\newcommand{\klab}[1]{\text{\scshape{K-#1}}}
\newcommand{\semlab}[1]{\text{\scshape{S-#1}}} \newcommand{\semlab}[1]{\text{\scshape{S-#1}}}
\newcommand{\tylab}[1]{\text{\scshape{T-#1}}} \newcommand{\tylab}[1]{\text{\scshape{T-#1}}}
\newcommand{\mlab}[1]{\text{\scshape{M-#1}}} \newcommand{\mlab}[1]{\text{\scshape{M-#1}}}
\newcommand{\siglab}[1]{\text{\scshape{Sig-#1}}} \newcommand{\siglab}[1]{\text{\scshape{Sig-#1}}}
\newcommand{\rowlab}[1]{\textrm{\scshape{R-#1}}} \newcommand{\rowlab}[1]{\text{\scshape{R-#1}}}
%% %%
%% Lindley's array stuff. %% Lindley's array stuff.

317
thesis.tex
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@@ -230,9 +230,8 @@ In \BCalc{}, types are precursory to terms, as it is intrinsically
typed. Thus we begin by presenting the syntax of its kind and type typed. Thus we begin by presenting the syntax of its kind and type
structure in Section~\ref{sec:base-language-types}. Subsequently in structure in Section~\ref{sec:base-language-types}. Subsequently in
Section~\ref{sec:base-language-terms} we present the term syntax, Section~\ref{sec:base-language-terms} we present the term syntax,
before presenting the formation rules for kinds and types in before presenting the formation rules for types in
Section~\ref{sec:base-language-kind-rules} and Section~\ref{sec:base-language-type-rules}.
Section~\ref{sec:base-language-type-rules}, respectively.
% Typically the presentation of a programming language begins with its % Typically the presentation of a programming language begins with its
% syntax. If the language is typed there are two possible starting % syntax. If the language is typed there are two possible starting
@@ -269,14 +268,15 @@ Section~\ref{sec:base-language-type-rules}, respectively.
\mid \Comp \mid \Effect \\ \mid \Comp \mid \Effect \\
\slab{Label sets} &\mathcal{L} &::=& \emptyset \mid \{\ell\} \uplus \mathcal{L}\\ \slab{Label sets} &\mathcal{L} &::=& \emptyset \mid \{\ell\} \uplus \mathcal{L}\\
%\slab{Type variables} &\alpha, \rho, \theta& \\ %\slab{Type variables} &\alpha, \rho, \theta& \\
% \slab{Type environments} &\Gamma &::=& \cdot \mid \Gamma, x:A \\ \slab{Type environments} &\Gamma &::=& \cdot \mid \Gamma, x:A \\
% \slab{Kind environments} &\Delta &::=& \cdot \mid \Delta, \alpha:K \slab{Kind environments} &\Delta &::=& \cdot \mid \Delta, \alpha:K
\end{syntax} \end{syntax}
\caption{Syntax of types in \BCalc{}.}\label{fig:base-language-types} \caption{Syntax of types, kinds, and their environments.}
\label{fig:base-language-types}
\end{figure} \end{figure}
% %
Figure~\ref{fig:base-language-types} depicts the syntax for kinds and Figure~\ref{fig:base-language-types} depicts the syntax for types,
types. kinds, and their environments.
% %
\paragraph{Value types} \paragraph{Value types}
We distinguish between values and computations at the level of We distinguish between values and computations at the level of
@@ -308,32 +308,31 @@ rather than $\rho$ and refer to it as an \emph{effect variable}).
% %
The row variable in an open row type can be instantiated with The row variable in an open row type can be instantiated with
additional labels. Each label may occur at most once in each row (we additional labels. Each label may occur at most once in each row (we
enforce this restriction at the level of kinds). Two rows are enforce this restriction at the level of kinds). We identify rows up
equivalent if they contain the same labels and both end in either to the reordering of labels.
$\cdot$ or a row variable $\rho$.
% %
\begin{mathpar} % \begin{mathpar}
\inferrule*[Lab=\rowlab{Closed}] % \inferrule*[Lab=\rowlab{Closed}]
{~} % {~}
{\cdot \equiv_{\mathrm{row}} \cdot} % {\cdot \equiv_{\mathrm{row}} \cdot}
\inferrule*[Lab=\rowlab{Open}] % \inferrule*[Lab=\rowlab{Open}]
{~} % {~}
{\rho \equiv_{\mathrm{row}} \rho} % {\rho \equiv_{\mathrm{row}} \rho}
\inferrule*[Lab=\rowlab{Head}] % \inferrule*[Lab=\rowlab{Head}]
{R \equiv_{\mathrm{row}} R'} % {R \equiv_{\mathrm{row}} R'}
{\ell:P;R \equiv_{\mathrm{row}} \ell:P;R'} % {\ell:P;R \equiv_{\mathrm{row}} \ell:P;R'}
\inferrule*[Lab=\rowlab{Swap}] % \inferrule*[Lab=\rowlab{Swap}]
{R \equiv_{\mathrm{row}} R'} % {R \equiv_{\mathrm{row}} R'}
{\ell:P;\ell':P';R \equiv_{\mathrm{row}} \ell':P';\ell:P;R'} % {\ell:P;\ell':P';R \equiv_{\mathrm{row}} \ell':P';\ell:P;R'}
\end{mathpar} % \end{mathpar}
% % %
The last rule $\rowlab{Swap}$ let us identify rows up to the % The last rule $\rowlab{Swap}$ let us identify rows up to the
reordering of labels. For instance, the two rows % reordering of labels. For instance, the two rows
$\ell_1 : P_1; \cdots; \ell_n : P_n; \cdot$ and % $\ell_1 : P_1; \cdots; \ell_n : P_n; \cdot$ and
$\ell_n : P_n; \cdots ; \ell_1 : P_1; \cdot$ are equivalent. % $\ell_n : P_n; \cdots ; \ell_1 : P_1; \cdot$ are equivalent.
% %
Absent labels in closed rows are redundant. Absent labels in closed rows are redundant.
% %
@@ -343,6 +342,97 @@ the zero type as the empty, closed variant $\ZeroType \defas
type, i.e. $\UnitType \defas \Record{\cdot}$. We shall often omit type, i.e. $\UnitType \defas \Record{\cdot}$. We shall often omit
$\cdot$ for closed rows. $\cdot$ for closed rows.
%
\begin{figure}
\begin{mathpar}
% alpha : K
\inferrule*[Lab=\klab{TyVar}]
{ }
{\Delta, \alpha : K \vdash \alpha : K}
% Computation
\inferrule*[Lab=\klab{Comp}]
{ \Delta \vdash A : \Type \\
\Delta \vdash E : \Effect \\
}
{\Delta \vdash A \eff E : \Comp}
% A -E-> B, A : Type, E : Row, B : Type
\inferrule*[Lab=\klab{Fun}]
{ \Delta \vdash A : \Type \\
\Delta \vdash C : \Comp \\
}
{\Delta \vdash A \to C : \Type}
% forall alpha : K . A : Type
\inferrule*[Lab=\klab{Forall}]
{ \Delta, \alpha : K \vdash C : \Comp}
{\Delta \vdash \forall \alpha^K . \, C : \Type}
% Record
\inferrule*[Lab=\klab{Record}]
{ \Delta \vdash R : \Row_\emptyset}
{\Delta \vdash \Record{R} : \Type}
% Variant
\inferrule*[Lab=\klab{Variant}]
{ \Delta \vdash R : \Row_\emptyset}
{\Delta \vdash [R] : \Type}
% Effect
\inferrule*[Lab=\klab{Effect}]
{ \Delta \vdash R : \Row_\emptyset}
{\Delta \vdash \{R\} : \Effect}
% Present
\inferrule*[Lab=\klab{Present}]
{\Delta \vdash A : \Type}
{\Delta \vdash \Pre{A} : \Presence}
% Absent
\inferrule*[Lab=\klab{Absent}]
{ }
{\Delta \vdash \Abs : \Presence}
% Empty row
\inferrule*[Lab=\klab{EmptyRow}]
{ }
{\Delta \vdash \cdot : \Row_\mathcal{L}}
% Extend row
\inferrule*[Lab=\klab{ExtendRow}]
{ \Delta \vdash P : \Presence \\
\Delta \vdash R : \Row_{\mathcal{L} \uplus \{\ell\}}
}
{\Delta \vdash \ell : P;R : \Row_\mathcal{L}}
\end{mathpar}
\caption{Kinding Rules}
\label{fig:base-language-kinding}
\end{figure}
%
\paragraph{Kinds}
The kinds comprise $\Type$ for regular type variables, $\Presence$ for
presence variables, $\Comp$ for computation type variables, $\Effect$
for effect variables, and lastly $\Row_{\mathcal{L}}$ for row
variables.
%
The formation rules for kinds are given in
Figure~\ref{fig:base-language-kinding}. The kinding judgement
$\Delta \vdash T : K$ states that type $T$ has kind $K$ in kind
environment $\Delta$.
%
The row kind is annotated by a set of labels $\mathcal{L}$. We use
this set to track the labels of a given row type to ensure uniqueness
amongst labels in each row type. For example, the kinding rule
$\klab{ExtendRow}$ uses this set to constrain which labels may be
mentioned in the tail of $R$. We shall elaborate on this in
Section~\ref{sec:row-polymorphism}.
\paragraph{Environments}
Kind and type environments are right-extended sequences of bindings. A
kind environment binds type variables to their kinds, whilst a type
environment binds term variables to their types.
% Value types comprise the function type $A \to C$, whose domain % Value types comprise the function type $A \to C$, whose domain
% is a value type and its codomain is a computation type $B \eff E$, % is a value type and its codomain is a computation type $B \eff E$,
@@ -430,25 +520,156 @@ kind information (term abstraction, type abstraction, injection,
operations, and empty cases). However, we shall omit these annotations operations, and empty cases). However, we shall omit these annotations
whenever they are clear from context. whenever they are clear from context.
\subsection{Kind and type environments}
\label{sec:kind-and-type-environments}
We define kind and type environments as right-extended sequences
mapping type variables to and their kinds and terms to their types,
respectively.
%
\begin{syntax}
\slab{Kind environments} &\Delta &::=& \cdot \mid \Delta, \alpha:K\\
\slab{Type environments} &\Gamma &::=& \cdot \mid \Gamma, x:A
\end{syntax}
%
Having introduced all the syntax, we now move onto introducing the
static semantics.
\subsection{Kinding rules}
\label{sec:base-language-kind-rules}
\subsection{Typing rules} \subsection{Typing rules}
\label{sec:base-language-type-rules} \label{sec:base-language-type-rules}
%
\begin{figure}
Values
\begin{mathpar}
% Variable
\inferrule*[Lab=\tylab{Var}]
{x : A \in \Gamma}
{\typv{\Delta;\Gamma}{x : A}}
% Abstraction
\inferrule*[Lab=\tylab{Lam}]
{\typ{\Delta;\Gamma, x : A}{M : C}}
{\typv{\Delta;\Gamma}{\lambda x^A .\, M : A \to C}}
% Polymorphic abstraction
\inferrule*[Lab=\tylab{PolyLam}]
{\typv{\Delta,\alpha : K;\Gamma}{M : C} \\
\alpha \notin FTV(\Gamma)
}
{\typv{\Delta;\Gamma}{\Lambda \alpha^K .\, M : \forall \alpha^K . \,C}}
\\
% unit : ()
\inferrule*[Lab=\tylab{Unit}]
{ }
{\typv{\Delta;\Gamma}{\Record{} : \Record{}}}
% Extension
\inferrule*[Lab=\tylab{Extend}]
{ \typv{\Delta;\Gamma}{V : A} \\
\typv{\Delta;\Gamma}{W : \Record{\ell:\Abs;R}}
}
{\typv{\Delta;\Gamma}{\Record{\ell=V;W} : \Record{\ell:\Pre{A};R}}}
% Inject
\inferrule*[Lab=\tylab{Inject}]
{\typv{\Delta;\Gamma}{V : A}}
{\typv{\Delta;\Gamma}{(\ell~V)^R : [\ell : \Pre{A}; R]}}
\end{mathpar}
Computations
\begin{mathpar}
% Application
\inferrule*[Lab=\tylab{App}]
{\typv{\Delta;\Gamma}{V : A \to C} \\
\typv{\Delta;\Gamma}{W : B}
}
{\typ{\Delta;\Gamma}{V\,W : C}}
% Polymorphic application
\inferrule*[Lab=\tylab{PolyApp}]
{\typv{\Delta;\Gamma}{V : \forall \alpha^K . \, C} \\
\Delta \vdash A : K
}
{\typ{\Delta;\Gamma}{V\,A : C[A/\alpha]}}
% Split
\inferrule*[Lab=\tylab{Split}]
{\typv{\Delta;\Gamma}{V : \Record{\ell : \Pre{A};R}} \\\\
\typ{\Delta;\Gamma, x : A, y : \Record{\ell : \Abs; R}}{N : C}
}
{\typ{\Delta;\Gamma}{\Let \; \Record{\ell =x;y} = V\; \In \; N : C}}
% Case
\inferrule*[Lab=\tylab{Case}]
{ \typv{\Delta;\Gamma}{V : [\ell : \Pre{A};R]} \\\\
\typ{\Delta;\Gamma,x:A}{M : C} \\\\
\typ{\Delta;\Gamma,y:[\ell : \Abs;R]}{N : C}
}
{\typ{\Delta;\Gamma}{\Case \; V \{\ell\; x \mapsto M;y \mapsto N \} : C}}
% Absurd
\inferrule*[Lab=\tylab{Absurd}]
{\typv{\Delta;\Gamma}{V : []}}
{\typ{\Delta;\Gamma}{\Absurd^C \; V : C}}
% Return
\inferrule*[Lab=\tylab{Return}]
{\typv{\Delta;\Gamma}{V : A}}
{\typc{\Delta;\Gamma}{\Return \; V : A}{E}}
\\
% Let
\inferrule*[Lab=\tylab{Let}]
{\typc{\Delta;\Gamma}{M : A}{E} \\
\typ{\Delta;\Gamma, x : A}{N : C}
}
{\typ{\Delta;\Gamma}{\Let \; x \revto M\; \In \; N : C}}
\end{mathpar}
\caption{Typing Rules}
\label{fig:base-language-type-rules}
\end{figure}
%
The typing rules are given by
Figure~\ref{fig:base-language-type-rules}. The \tylab{Var} rule states
that a variable $x$ has type $A$ if $x$ is bound to $A$ in the type
environment $\Gamma$. The \tylab{Lam} rules states that a lambda value
$(\lambda x.M)$ has type $A \to C$ if the computation $M$ has type $C$
assuming $x : A$. In the \tylab{PolyLam} rule we make use of the set
of \emph{free type variables} (FTV).
%
We define FTV by mutual induction over type environments, $\Gamma$,
and the type structure:
%
\[
\ba[t]{@{~}l@{~~~~~~}c@{~}l}
\begin{eqs}
\FTV(\alpha) &\defas& \{\alpha\}\\
\FTV(\forall \alpha^K.C) &\defas& \FTV(C) \setminus \{\alpha\}\\
\FTV(A \to C) &\defas& \FTV(A) \cup \FTV(C)\\
\FTV(A \eff E) &\defas& \FTV(A) \cup \FTV(E)\\
% \FTV(\{R\}) &\defas& \FTV(R)\\
% \FTV(\Record{R}) &\defas& \FTV(R)\\
% \FTV([R]) &\defas& \FTV(R)\\
% \FTV(l:P;R) &\defas& \FTV(P) \cup \FTV(R)\\
% \FTV(\Pre{A}) &\defas& \FTV(A)\\
% \FTV(\Abs) &\defas& \emptyset\\
% \FTV(\theta) &\defas& \{\theta\}
\end{eqs} & &
\begin{eqs}
\FTV([R]) &\defas& \FTV(R)\\
\FTV(\Record{R}) &\defas& \FTV(R)\\
\FTV(\{R\}) &\defas& \FTV(R)\\
\FTV(l:P;R) &\defas& \FTV(P) \cup \FTV(R)\\
\end{eqs}\\\\
\begin{eqs}
\FTV(\theta) &\defas& \{\theta\}\\
\FTV(\Abs) &\defas& \emptyset\\
\FTV(\Pre{A}) &\defas& \FTV(A)
\end{eqs} & &
\begin{eqs}
\FTV(\cdot) &\defas& \emptyset\\
\FTV(\Gamma,x : A) &\defas& \FTV(\Gamma) \cup \FTV(A)
\end{eqs}
\ea
\]
%
Thus the rule states that a type abstraction $(\Lambda \alpha. M)$ has
type $\forall \alpha.C$ if the computation $M$ has type $C$ assuming
$\alpha : K$ and $\alpha$ does not appear in the free type variables
of current type environment $\Gamma$. The \tylab{Unit} rule provides
the basis for all records as it simply states that the empty record
has type unit. The \tylab{Extend} rule handles record
extension. Supposing we wish to extend some record $\Record{W}$ with
$\ell = V$, that is $\Record{\ell = V; W}$. This extension has type
$\Record{\ell : \Pre{A};R}$ if and only if $V$ is well-typed and we
can ascribe $W : \Record{\ell : \Abs; R}$. In other words, the label
$\ell$ must not be mentioned in $W$. The \tylab{Inject} rule states
that $(\ell~V)^R$ has type $[\ell : \Pre{A}; R]$ if the payload $V$ is
well-typed.
\section{Dynamic semantics} \section{Dynamic semantics}
@@ -469,6 +690,8 @@ static semantics.
\subsection{Effect inference} \subsection{Effect inference}
\subsection{Dynamic semantics} \subsection{Dynamic semantics}
\section{Default handlers}
\section{Parameterised handlers} \section{Parameterised handlers}
\section{Shallow handlers} \section{Shallow handlers}