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Syntactic categories.

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Daniel Hillerström 6 years ago
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  1. 14
      macros.tex
  2. 137
      thesis.tex

14
macros.tex
View File

@ -105,10 +105,18 @@
\newcommand{\CompCat}{\CatName{Comp}}
\newcommand{\ValCat}{\CatName{Val}}
\newcommand{\VarCat}{\CatName{Var}}
\newcommand{\TypCat}{\CatName{Type}}
\newcommand{\TyVarCat}{\CatName{TyVar}}
\newcommand{\ValTypeCat}{\CatName{TVal}}
\newcommand{\CompTypeCat}{\CatName{TComp}}
\newcommand{\PresenceCat}{\CatName{Presence}}
\newcommand{\TypeCat}{\CatName{Type}}
\newcommand{\TyVarCat}{\CatName{TVar}}
\newcommand{\KindCat}{\CatName{Kind}}
\newcommand{\RowCat}{\RowCat}
\newcommand{\RowCat}{\CatName{Row}}
\newcommand{\EffectCat}{\CatName{Effect}}
\newcommand{\TermCat}{\CatName{Term}}
\newcommand{\LabelCat}{\CatName{Label}}
\newcommand{\TyEnvCat}{\CatName{TyEnv}}
\newcommand{\KindEnvCat}{\CatName{KindEnv}}
%%
%% Lindley's array stuff.

137
thesis.tex
View File

@ -251,26 +251,42 @@ Section~\ref{sec:base-language-type-rules}.
\label{sec:base-language-types}
%
\begin{figure}
\begin{syntax}
\slab{Value types} &A,B &::= & A \to C
\mid \alpha
\begin{syntax}
% \slab{Value types} &A,B &::= & A \to C
% \mid \alpha
% \mid \forall \alpha^K.C
% \mid \Record{R}
% \mid [R]\\
% \slab{Computation types}
% &C,D &::= & A \eff E \\
% \slab{Effect types} &E &::= & \{R\}\\
% \slab{Row types} &R &::= & \ell : P;R \mid \rho \mid \cdot \\
% \slab{Presence types} &P &::= & \Pre{A} \mid \Abs \mid \theta\\
% %\slab{Labels} &\ell & & \\
% % \slab{Types} &T &::= & A \mid C \mid E \mid R \mid P \\
% \slab{Kinds} &K &::= & \Type \mid \Row_\mathcal{L} \mid \Presence
% \mid \Comp \mid \Effect \\
% \slab{Label sets} &\mathcal{L} &::=& \emptyset \mid \{\ell\} \uplus \mathcal{L}\\
% %\slab{Type variables} &\alpha, \rho, \theta& \\
% \slab{Type environments} &\Gamma &::=& \cdot \mid \Gamma, x:A \\
% \slab{Kind environments} &\Delta &::=& \cdot \mid \Delta, \alpha:K
\slab{Value types} &A,B \in \ValTypeCat &::= & A \to C
\mid \forall \alpha^K.C
\mid \Record{R}
\mid [R]\\
\slab{Computation types}
&C,D &::= & A \eff E \\
\slab{Effect types} &E &::= & \{R\}\\
\slab{Row types} &R &::= & \ell : P;R \mid \rho \mid \cdot \\
\slab{Presence types} &P &::= & \Pre{A} \mid \Abs \mid \theta\\
%\slab{Labels} &\ell & & \\
% \slab{Types} &T &::= & A \mid C \mid E \mid R \mid P \\
\slab{Kinds} &K &::= & \Type \mid \Row_\mathcal{L} \mid \Presence
\mid \Comp \mid \Effect \\
\slab{Label sets} &\mathcal{L} &::=& \emptyset \mid \{\ell\} \uplus \mathcal{L}\\
%\slab{Type variables} &\alpha, \rho, \theta& \\
\slab{Type environments} &\Gamma &::=& \cdot \mid \Gamma, x:A \\
\slab{Kind environments} &\Delta &::=& \cdot \mid \Delta, \alpha:K
\end{syntax}
\mid \Record{R} \mid [R]
\mid \alpha \\
\slab{Computation types\!\!}
&C,D \in \CompTypeCat &::= & A \eff E \\
\slab{Effect types} &E \in \EffectCat &::= & \{R\}\\
\slab{Row types} &R \in \RowCat &::= & \ell : P;R \mid \rho \mid \cdot \\
\slab{Presence types\!\!\!\!\!} &P \in \PresenceCat &::= & \Pre{A} \mid \Abs \mid \theta\\
\\
\slab{Types} &T \in \TypeCat &::= & A \mid C \mid E \mid R \mid P \\
\slab{Kinds} &K \in \KindCat &::= & \Type \mid \Comp \mid \Effect \mid \Row_\mathcal{L} \mid \Presence \\
\slab{Label sets} &\mathcal{L} \in \LabelCat &::=& \emptyset \mid \{\ell\} \uplus \mathcal{L}\\\\
\slab{Type environments} &\Gamma \in \TyEnvCat &::=& \cdot \mid \Gamma, x:A \\
\slab{Kind environments} &\Delta \in \KindEnvCat &::=& \cdot \mid \Delta, \alpha:K \\
\end{syntax}
\caption{Syntax of types, kinds, and their environments.}
\label{fig:base-language-types}
\end{figure}
@ -434,6 +450,21 @@ 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.
\paragraph{Type variable} The type structure has three syntactically
distinct type variables (the kinding system gives us five semantically
distinct notions of type variables). As we sometimes wish to refer
collectively to type variables, we define the set of type variables,
$\TyVarCat$, to be generated by:
%
\[
\TyVarCat \defas
\ba[t]{@{~}l@{~}l}
&\{ A \in \ValTypeCat \mid A \text{ has the form } \alpha \}\\
\cup &\{ R \in \RowCat \mid R \text{ has the form } \rho \}\\
\cup &\{ P \in \PresenceCat \mid P \text{ has the form } \theta \}
\ea
\]
% Value types comprise the function type $A \to C$, whose domain
% is a value type and its codomain is a computation type $B \eff E$,
% where $E$ is an effect type detailing which effects the implementing
@ -472,7 +503,8 @@ environment binds term variables to their types.
\slab{Computations} &M,N \in \CompCat &::= & V\,W \mid V\,A\\
& &\mid& \Let\; \Record{\ell=x;y} = V \; \In \; N\\
& &\mid& \Case\; V \{\ell~x \mapsto M; y \mapsto N\} \mid \Absurd^C~V\\
& &\mid& \Return~V \mid \Let \; x \revto M \; \In \; N
& &\mid& \Return~V \mid \Let \; x \revto M \; \In \; N\\
\slab{Terms} &T \in \TermCat &::= & x \mid V \mid M
\end{syntax}
\caption{Term syntax of \BCalc{}.}
@ -539,7 +571,7 @@ Values
% Polymorphic abstraction
\inferrule*[Lab=\tylab{PolyLam}]
{\typv{\Delta,\alpha : K;\Gamma}{M : C} \\
\alpha \notin FTV(\Gamma)
\alpha \notin \FTV(\Gamma)
}
{\typv{\Delta;\Gamma}{\Lambda \alpha^K .\, M : \forall \alpha^K . \,C}}
\\
@ -619,46 +651,61 @@ Figure~\ref{fig:base-language-type-rules}. In each typing rule, we
implicitly assume that each type is well-kinded with respect to the
kinding environment $\Delta$.
\paragraph{Typing values} 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).
\paragraph{Typing values} 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} ($\FTV$) to ensure that we do not
inadvertently capture a free type variable from the context.
%
We define FTV by mutual induction over type environments, $\Gamma$,
and the type structure:
We define $\FTV$ by mutual induction over type environments, $\Gamma$,
and the type structure, $T$, as follows.
%
\[
\ba[t]{@{~}l@{~~~~~~}c@{~}l}
\begin{eqs}
\ba[t]{@{}l}
\begin{eqs}
\FTV &:& \ValTypeCat \to \TyVarCat\\
\FTV(\alpha) &\defas& \{\alpha\}\\
\FTV(\forall \alpha^K.C) &\defas& \FTV(C) \setminus \{\alpha\}\\
\FTV(A \to C) &\defas& \FTV(A) \cup \FTV(C)\\
\FTV(A \eff E) &\defas& \FTV(A) \cup \FTV(E)\\
% \FTV(\{R\}) &\defas& \FTV(R)\\
% \FTV(\Record{R}) &\defas& \FTV(R)\\
% \FTV([R]) &\defas& \FTV(R)\\
\FTV(\{R\}) &\defas& \FTV(R)\\
\FTV(\Record{R}) &\defas& \FTV(R)\\
\FTV([R]) &\defas& \FTV(R)\\
% \FTV(l:P;R) &\defas& \FTV(P) \cup \FTV(R)\\
% \FTV(\Pre{A}) &\defas& \FTV(A)\\
% \FTV(\Abs) &\defas& \emptyset\\
% \FTV(\theta) &\defas& \{\theta\}
\end{eqs} & &
\end{eqs}\ea & &
\begin{eqs}
\FTV([R]) &\defas& \FTV(R)\\
\FTV(\Record{R}) &\defas& \FTV(R)\\
\FTV(\{R\}) &\defas& \FTV(R)\\
% \FTV([R]) &\defas& \FTV(R)\\
% \FTV(\Record{R}) &\defas& \FTV(R)\\
% \FTV(\{R\}) &\defas& \FTV(R)\\
\FTV &:& \RowCat \to \TyVarCat\\
\FTV(l:P;R) &\defas& \FTV(P) \cup \FTV(R)\\
\end{eqs}\\\\
\begin{eqs}
\FTV(\cdot) &\defas& \emptyset\\\\
\FTV &:& \PresenceCat \to \TyVarCat\\
\FTV(\theta) &\defas& \{\theta\}\\
\FTV(\Abs) &\defas& \emptyset\\
\FTV(\Pre{A}) &\defas& \FTV(A)
\end{eqs} & &
\begin{eqs}
\FTV(\Pre{A}) &\defas& \FTV(A)\\
\end{eqs}\\\\
\multicolumn{3}{c}{\begin{eqs}
\FTV &:& \TyEnvCat \to \TyVarCat\\
\FTV(\cdot) &\defas& \emptyset\\
\FTV(\Gamma,x : A) &\defas& \FTV(\Gamma) \cup \FTV(A)
\end{eqs}
\end{eqs}}
% \begin{eqs}
% \FTV(\theta) &\defas& \{\theta\}\\
% \FTV(\Abs) &\defas& \emptyset\\
% \FTV(\Pre{A}) &\defas& \FTV(A)
% \end{eqs} & &
% \begin{eqs}
% \FTV(\cdot) &\defas& \emptyset\\
% \FTV(\Gamma,x : A) &\defas& \FTV(\Gamma) \cup \FTV(A)
% \end{eqs}
\ea
\]
%
@ -691,10 +738,12 @@ application $V\,A$ is well-typed whenever the abstractor term $V$ has
the polymorphic type $\forall \alpha^K.C$ and the type $A$ has kind
$K$. This rule makes use of type substitution. We write $C[A/\alpha]$
to mean substitute some type $A$ for some type variable $\alpha$ in
some type $C$. We define type substitution as a ternary function defined as follows
some type $C$. We define type substitution as a ternary function
defined as follows.
%
\[
\begin{eqs}
-[-/-] &:& \TypeCat \times \TypeCat \times \TyVarCat \to \TypeCat\\
(A \to C)[B/\alpha] &\defas& A[B/\alpha] \to C[B/\alpha]\\
\alpha[B/\beta] &\defas& \begin{cases}
B & \text{if } \alpha = \beta\\

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