Add ANF transformation. Define LetFJ. Change Constraint generation. Cleanup old explanation of Capture conversion. Add example to Constraint generation
This commit is contained in:
Andreas Stadelmeier
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289
constraints.tex
289
constraints.tex
@ -9,19 +9,22 @@
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%\subsection{Well-Formedness}
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Constraint generation step is the same as in \cite{TIforFGJ} except for field access and method invocation.
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Here \fjtype{} generates capture constraints instead of normal subtype constraints.
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Subtype constraints are created according to the type rules defined in section \ref{sec:tifj}.
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The \typeExpr{} function creates constraints for a given expression.
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% But it can be easily adapted to Featherweight Java or Java.
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% We add T <. a for every return of an expression anyway. If anything returns a Generic like X it is not directly used in a method call like X <c T
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$\begin{array}{c}
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\typeExpr(\texttt{t}_1, \ntv{b}) = C_l \quad \quad \typeExpr(\texttt{t}_2, \ntv{c}) = C_l \quad \quad \ntv{b}, \ntv{c} \ \text{fresh}
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\\
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\hline
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\typeExpr(\texttt{t}_1 \texttt{?:} \texttt{t}_2, \tv{a}) = C_l \cup C_r \cup \set{\ntv{b} \lessdot \tv{a}, \ntv{c} \lessdot \tv{a}}
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\end{array}
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$
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The constraint generation works on the \TamedFJ{} language.
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This step is mostly same as in \cite{TIforFGJ} except for field access and method invocation.
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We will focus on those parts.
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Here the new capture constraints and wildcard type placeholders are introduced.
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Generally subtype constraints for an expression mirror the subtype relations in the premise of the respective type rule introduced in section \ref{sec:tifj}
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Unknown types at the time of the constraint generation step are replaced with type placeholders.
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\begin{verbatim}
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m(l, v){
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let x = x in x.add(v)
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}
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\end{verbatim}
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The constraint generation step cannot determine if a capture conversion is needed for a field access or a method call.
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Those statements produce $\lessdotCC$ constraints which signal the \unify{} algorithm that they qualify for a capture conversion.
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@ -68,7 +71,7 @@ For the example above a correct solution would be $\sigma(\tv{a}) = \wctype{\rwi
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\begin{align*}
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% Type
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\type{T}, \type{U} &::= \tv{a} \mid \wtv{a} \mid \mv{X} \mid {\wcNtype{\Delta}{N}} && \text{types and type placeholders}\\
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\type{N} &::= \exptype{C}{\il{T}} && \text{class type (with type variables)} \\
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\type{N} &::= \exptype{C}{\ol{T}} && \text{class type (with type variables)} \\
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% Constraints
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\constraint &::= \type{T} \lessdot \type{U} \mid \type{T} \lessdotCC \type{U} && \text{Constraint}\\
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\consSet &::= \set{\constraints} && \text{constraint set}\\
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@ -90,21 +93,17 @@ For the example above a correct solution would be $\sigma(\tv{a}) = \wctype{\rwi
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\begin{figure}[tp]
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\begin{gather*}
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\begin{array}{@{}l@{}l}
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\fjtype & ({\mtypeEnvironment}, \mathtt{class } \ \exptype{C}{\ol{X} \triangleleft \ol{N}} \ \mathtt{ extends } \ \mathtt{N \{ \overline{T} \ \overline{f}; \, K \, \overline{M} \}}) =\\
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\fjtype & ({\mtypeEnvironment}, \mathtt{class } \ \exptype{C}{\ol{X} \triangleleft \ol{N}} \ \mathtt{ extends } \ \mathtt{N \{ \overline{T} \ \overline{f}; \, \overline{M} \}}) =\\
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& \begin{array}{ll@{}l}
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\textbf{let} & \forall \texttt{m} \in \ol{M}: \tv{a}_\texttt{m}, \ol{\tv{a}_m} \ \text{fresh} \\
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& \ol{\methodAssumption} = \begin{array}[t]{l}
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\set{ \mv{m'} : (\exptype{C}{\ol{X} \triangleleft \ol{N}} \to \ol{\tv{a}} \to \tv{a}) \mid
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\mv{m'} \in \ol{M} \setminus \set{\mv{m}}, \, \tv{a}\, \ol{\tv{a}}\ \text{fresh} } \\
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\ \cup \, \set{\mv{m} : (\exptype{C}{\ol{X} \triangleleft \ol{N}} \to \ol{\tv{a}_m} \to \tv{a}_\mv{m})}
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\end{array}
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\\
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& C_m = \typeExpr(\mtypeEnvironment \cup \set{\mv{this} :
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\exptype{C}{\ol{X}} , \, \ol{x} : \ol{\tv{a}_m} }, \texttt{e}, \tv{a}_\texttt{m}) \\
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\textbf{let} & \ol{\methodAssumption} =
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\set{ \mv{m} : (\exptype{C}{\ol{X}}, \ol{\tv{a}} \to \tv{a}) \mid
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\set{ \mv{m}(\ol{x}) = \expr{e} } \in \ol{M}, \, \tv{a}, \ol{\tv{a}}\ \text{fresh} } \\
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\textbf{in}
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& { ( \mtypeEnvironment \cup \ol{\methodAssumption}, \,
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\bigcup_{\texttt{m} \in \ol{M}} C_m )
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}
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& \begin{array}[t]{l}
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\set{ \typeExpr(\mtypeEnvironment \cup \ol{\methodAssumption} \cup \set{\mv{this} :
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\exptype{C}{\ol{X}} , \, \ol{x} : \ol{\tv{a}} }, \texttt{e}, \tv{a})
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\\ \quad \quad \quad \quad \mid \set{ \mv{m}(\ol{x}) = \expr{e} } \in \ol{M},\, \mv{m} : (\exptype{C}{\ol{X}}, \ol{\tv{a}} \to \tv{a}) \in \ol{\methodAssumption}}
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\end{array}
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\end{array}
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\end{array}
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\end{gather*}
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@ -112,56 +111,6 @@ For the example above a correct solution would be $\sigma(\tv{a}) = \wctype{\rwi
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\label{fig:constraints-for-classes}
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\end{figure}
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% \textbf{Method Assumptions}
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% %$\Pi$ is a set of method assumptions used by the $\fjtype{}$ algorithm.
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% % \begin{verbatim}
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% % class Example<X> {
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% % <Y> Y m(Example<Y> p){ ... }
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% % }
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% % \end{verbatim}
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% In Featherweight Java a method type is bound to a specific class.
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% The class \texttt{Example} shown above contains one method \texttt{m}:
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% \begin{displaymath}
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% \textit{mtype}({\texttt{m}, \exptype{Example}{\type{X}}}) = \generics{\type{Y}} \ \exptype{Example}{\type{Y}} \to \type{Y}
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% \end{displaymath}
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% $\Pi$ is a set of method assumptions used by the $\fjtype{}$ algorithm.
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% It's a map of method types to method names.
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% Every method name has a set of possible types,
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% because there could be more than one method with the same name in a program consisting out of multiple classes.
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% To simplify the syntax of method assumptions, we add the inheriting class type to the parameter list:
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% \begin{displaymath}
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% \Pi = \set{ \texttt{m} : \generics{\type{X}, \type{Y}} \ (\exptype{Example}{\type{X}}, \exptype{Example}{\type{Y}}) \to \type{Y}}
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% \end{displaymath}
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% \begin{verbatim}
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% class Example<X> { }
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% <X, Y> Y m(Example<X> this, Example<Y> p){ ... }
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% \end{verbatim}
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\begin{displaymath}
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\begin{array}{@{}l@{}l}
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\typeExpr{} &({\mtypeEnvironment} , \texttt{let}\ \expr{x} = \expr{e}_1 \ \texttt{in} \ \expr{e}_2) = \\
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& \begin{array}{ll}
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\textbf{let}
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& \tv{e}, \tv{x} \ \text{fresh} \\
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& \consSet_R = \typeExpr({\mtypeEnvironment}, \expr{e}_1, \tv{e})\\
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& \constraint =
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\set{
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\tv{e} \lessdot \tv{x}
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}\\
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{\mathbf{in}} & {
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\consSet_R \cup \set{\constraint}} \cup \typeExpr(\mtypeEnvironment \cup \set{\expr{x} : \tv{x}})
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\end{array}
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\end{array}
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\end{displaymath}
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\begin{displaymath}
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\begin{array}{@{}l@{}l}
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\typeExpr{} &({\mtypeEnvironment} , \texttt{e}.\texttt{f}, \tv{a}) = \\
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@ -185,13 +134,27 @@ For the example above a correct solution would be $\sigma(\tv{a}) = \wctype{\rwi
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\end{array}
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\end{displaymath}
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The set of method assumptions returned by the \textit{mtypes} function is used to generate the constraints for a method call expression:
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There are two kinds of method calls.
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The ones to already typed methods and calls to untyped methods.
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\begin{displaymath}
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\begin{array}{@{}l@{}l}
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\typeExpr{}' & ({\mtypeEnvironment} , \expr{v}.\mathtt{m}(\overline{\expr{v}}), \tv{a}) = \\
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\typeExpr{} &({\mtypeEnvironment} , \texttt{let}\ \expr{x} = \expr{e}_1 \ \texttt{in} \ \expr{e}_2, \tv{a}) = \\
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& \begin{array}{ll}
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\textbf{let}
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& \tv{e}_1, \tv{e}_2, \tv{x} \ \text{fresh} \\
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& \consSet_1 = \typeExpr({\mtypeEnvironment}, \expr{e}_1, \tv{e}_1)\\
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& \consSet_2 = \typeExpr({\mtypeEnvironment} \cup \set{\expr{x} : \tv{x}}, \expr{e}_2, \tv{e}_2)\\
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& \constraint =
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\set{
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\tv{e}_1 \lessdot \tv{x}, \tv{e}_2 \lessdot \tv{a}
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}\\
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{\mathbf{in}} & {
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\consSet_1 \cup \consSet_2 \cup \set{\constraint}}
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\end{array}
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\end{array}
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\end{displaymath}
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\begin{displaymath}
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\begin{array}{@{}l@{}l}
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\typeExpr{} & ({\mtypeEnvironment} , \expr{v}.\mathtt{m}(\overline{\expr{v}}), \tv{a}) = \\
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& \begin{array}{ll}
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\textbf{let}
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& \tv{r}, \ol{\tv{r}} \text{ fresh} \\
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@ -207,126 +170,98 @@ The ones to already typed methods and calls to untyped methods.
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\end{array}
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\end{array}
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\end{displaymath}
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\begin{displaymath}
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\begin{array}{@{}l@{}l}
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\typeExpr{}' & ({\mtypeEnvironment} , \texttt{e}.\mathtt{m}(\overline{\texttt{e}}), \tv{a}) = \\
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& \begin{array}{ll}
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\textbf{let}
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& \tv{r}, \ol{\tv{r}} \text{ fresh} \\
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& \consSet_R = \typeExpr(({\mtypeEnvironment} ;
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\overline{\localVarAssumption}), \texttt{e}, \tv{r})\\
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& \overline{\consSet} = \typeExpr({\mtypeEnvironment}, \ol{e}, \ol{\tv{r}}) \\
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& \begin{array}{@{}l@{}l}
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\constraint = \orCons\set{ &
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\begin{array}[t]{l}
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[\overline{\wtv{a}}/\ol{X}] [\overline{\wtv{b}}/\ol{Y}] \{ \tv{r} \lessdotCC \exptype{C}{\ol{X}},
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\overline{\tv{r}} \lessdotCC \ol{T}, \type{T} \lessdot \tv{a},
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\ol{Y} \lessdot \ol{N} \}
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\end{array}\\
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& \ |\
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(\texttt{m} : \generics{\ol{Y} \triangleleft \ol{N}}\overline{\type{T}} \to \type{T}) \in
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{\mtypeEnvironment} %, \, |\ol{T}| = |\ol{e}|
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, \, \overline{\wtv{a}} \text{ fresh}}
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\end{array}\\
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\mathbf{in} & (\consSet_R \cup \overline{\consSet} \cup \constraint, \type{T})
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\end{array}
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\end{array}
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\end{displaymath}
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\\[1em]
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\noindent
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\textbf{Example:}
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\begin{verbatim}
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class Class1{
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<X> X m(List<X> lx, List<? extends X> lt){ ... }
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List<? extends String> wGet(){ ... }
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List<String> get() { ... }
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<A> A head(List<X> l){ ... }
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List<? extends String> get() { ... }
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}
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class Class2{
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example(c1){
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return c1.m(c1.get(), c1.wGet());
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return c1.head(c1.get());
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}
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}
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\end{verbatim}
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%This example comes with predefined type annotations.
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We assume the class \texttt{Class1} has already been processed by our type inference algorithm,
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which has lead to the given type annotations for \texttt{Class1}.
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Now we call the $\fjtype{}$ function with the class \texttt{Class2} and the method assumptions for the preceeding class:
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We assume the class \texttt{Class1} has already been processed by our type inference algorithm
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leading to the following type annotations:
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%Now we call the $\fjtype{}$ function with the class \texttt{Class2} and the method assumptions for the preceeding class:
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\begin{displaymath}
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\mtypeEnvironment = \left\{\begin{array}{l}
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\type{Class1}.\texttt{m}: \generics{\type{X} \triangleleft \type{Object}} \
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(\exptype{List}{\type{X}}, \, \wctype{\wildcard{A}{\type{X}}{\bot}}{List}{\wildcard{A}{\type{X}}{\bot}}) \to \type{X}, \\
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\type{Class1}.\texttt{wGet}: () \to \wctype{\wildcard{A}{\type{Object}}{\type{String}}}{List}{\wildcard{A}{\type{Object}}{\type{String}}}, \\
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\type{Class1}.\texttt{get}: () \to \exptype{List}{\type{String}}
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\texttt{m}: \generics{\type{A} \triangleleft \type{Object}} \
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(\type{Class1},\, \exptype{List}{\type{A}}) \to \type{X}, \\
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\texttt{get}: (\type{Class1}) \to \wctype{\wildcard{A}{\type{Object}}{\type{String}}}{List}{\rwildcard{A}}
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\end{array} \right\}
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\end{displaymath}
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The result of the $\typeExpr{}$ function is the constraint set
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\begin{displaymath}
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C = \left\{ \begin{array}{l}
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\tv{c1} \lessdot \type{Class1}, \\
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\tv{p1} \lessdot \exptype{List}{\wtv{x}}, \\
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\exptype{List}{\type{String}} \lessdot \tv{p1}, \\
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\tv{p2} \lessdot \wctype{\wildcard{A}{\wtv{x}}{\bot}}{List}{\rwildcard{A}}, \\
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\wctype{\wildcard{A}{\type{String}}{\bot}}{List}{\rwildcard{A}} \lessdot \tv{p2}
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\end{array} \right\}
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\end{displaymath}
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At first we have to convert the example method to a syntactically correct \TamedFJ{} program.
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Afterwards the the \fjtype{} algorithm is able to generate constraints.
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\begin{minipage}{0.45\textwidth}
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\begin{lstlisting}[style=tamedfj]
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class Class2 {
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example(c1) = let x = c1 in
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let xp = x.get() in x.m(xp);
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}
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\end{lstlisting}
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\end{minipage}%
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\hfill
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\begin{minipage}{0.5\textwidth}
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\begin{constraintset}
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$
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\begin{array}{l}
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\ntv{c1} \lessdot \ntv{x}, \ntv{x} \lessdotCC \type{Class1}, \\
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\ntv{c1} \lessdot \ntv{x}, \ntv{x} \lessdotCC \type{Class1}, \\
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\wctype{\wildcard{A}{\type{String}}{\bot}}{List}{\rwildcard{A}} \lessdot \tv{xp}, \\
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\tv{xp} \lessdotCC \exptype{List}{\wtv{a}}
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\end{array}
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$
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\end{constraintset}
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\end{minipage}
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The first parameter of a method assumption is the receiver type $\type{T}_r$.
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\texttt{Class1} for this example.
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Therefore the $(\tv{c1} \lessdot \type{Class1})$ constraint.
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The type variable $\tv{c1}$ is assigned to the parameter \texttt{c1} of the \texttt{example} method.
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Following is a possible solution for the given constraint set:
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or a simplified version:
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\begin{minipage}{0.55\textwidth}
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\begin{lstlisting}[style=letfj]
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class Class2 {
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example(c1) = let x : Class1 = c1 in
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let xp : (*@$\wctype{\wildcard{A}{\type{String}}{\bot}}{List}{\rwildcard{A}}$@*) = x.get()
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in x.m(xp);
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}
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\end{lstlisting}
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\end{minipage}%
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\hfill
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\begin{minipage}{0.4\textwidth}
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\begin{constraintset}
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$
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\begin{array}{l}
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\sigma(\ntv{x}) = \type{Class1} \\
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%\tv{xp} \lessdot \exptype{List}{\wtv{x}}, \\
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%\exptype{List}{\type{String}} \lessdot \tv{p1}, \\
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\sigma(\tv{xp}) = \wctype{\wildcard{A}{\type{String}}{\bot}}{List}{\rwildcard{A}} \\
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\end{array}
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$
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\end{constraintset}
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\end{minipage}
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\begin{displaymath}
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C = \left\{ \begin{array}{l}
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\tv{c1} \lessdot \type{Class1}, \\
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\exptype{List}{\type{String}}
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\lessdot \exptype{List}{\wtv{x}}, \\
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\wctype{\wildcard{A}{\type{String}}{\bot}}{List}{\rwildcard{A}}
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\lessdot \wctype{\wildcard{A}{\wtv{x}}{\bot}}{List}{\rwildcard{A}}
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\end{array} \right\}
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\end{displaymath}
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For $\wctype{\wildcard{A}{\type{String}}{\bot}}{List}{\rwildcard{A}}$ to be a correct solution for $\tv{xp}$
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the constraint $\wctype{\wildcard{A}{\type{String}}{\bot}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\wtv{a}}$
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must be satisfied.
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This is possible, because we deal with a capture constraint.
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The $\lessdotCC$ constraint allows the left side to undergo a capture conversion
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which leads to $\exptype{List}{\rwildcard{A}} \lessdot \exptype{List}{\wtv{a}}$.
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Now a substitution of the wildcard placeholder $\wtv{a}$ with $\rwildcard{A}$ leads to a satisfied constraint set.
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$\wtv{x}$ is a type variable we use for the generic $\type{X}$. It is flagged as a free type variable.
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%TODO: Do an example where wildcards are inserted and we need capture conversion
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\unify{} returns the solution $(\sigma = \set{ \tv{x} \to \type{String} })$
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% \\[1em]
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% \noindent
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% \textbf{Example:}
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% \begin{verbatim}
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% class Class1 {
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% <X> Pair<X, X> make(List<X> l){ ... }
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% <Y> boolean compare(Pair<Y,Y> p) { ... }
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% example(l){
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% return compare(make(l));
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% }
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% }
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% \end{verbatim}
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% The method call \texttt{make(l)} generates the constraints
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% \begin{displaymath}
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% \tv{l} \lessdot \exptype{List}{\tv{x}}, \exptype{Pair}{\tv{x}, \tv{x}} \lessdot \tv{m}
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% \end{displaymath}
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% with $\tv{l}$ being the type placeholder for the variable \texttt{l}
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% and $\tv{m}$ is the type variable for the return type of the method call.
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% $\tv{m}$ is then used as the parameter for the \texttt{compare} method:
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% \begin{displaymath}
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% \tv{m} \lessdot \exptype{Pair}{\tv{y}, \tv{y}}
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% \end{displaymath}
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% Note the conversion of the generic parameters \texttt{X} and \texttt{Y} to type variables $\tv{x}$ and $\tv{y}$.
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|
||||
% Step 3 of the \unify{} algorithm has to look for every possible supertype of $\exptype{Pair}{\tv{x}, \tv{x}}$,
|
||||
% when processing the $\exptype{Pair}{\tv{x}, \tv{x}} \lessdot \tv{m}$ constraint.
|
||||
The wildcard placeholders are not used as parameter or return types of methods.
|
||||
Or as types for variables introduced by let statements.
|
||||
They are only used for generic method parameters during a method invocation.
|
||||
Type placeholders which are not flagged as wildcard placeholders ($\wtv{a}$) can never hold a free variable or a type containing free variables.
|
||||
This practice hinders free variables to leave their scope.
|
||||
The free variable $\rwildcard{A}$ generated by the capture conversion on the type $\wctype{\wildcard{A}{\type{String}}{\bot}}{List}{\rwildcard{A}}$
|
||||
cannot be used anywhere else then inside the constraints generated by the method call \texttt{x.m(xp)}.
|
||||
|
||||
\begin{displaymath}
|
||||
\begin{array}{@{}l@{}l}
|
||||
|
||||
322
introduction.tex
322
introduction.tex
@ -382,6 +382,91 @@ $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}} \lessdotCC \exptype
|
||||
% List<?> m() = new List<String>() :? new List<Integer>() :? id(m());
|
||||
% \end{verbatim}
|
||||
|
||||
\subsection{\TamedFJ{} and \letfj{}}
|
||||
%LetFJ -> Output language!
|
||||
%TamedFJ -> ANF transformed input langauge
|
||||
%Input language only described here. It is standard Featherweight Java
|
||||
% we use the transformation to proof soundness. this could also be moved to the end.
|
||||
% the constraint generation step assumes every method argument to be encapsulated in a let statement. This is the way Java is doing capture conversion
|
||||
|
||||
The input to our algorithm is a typeless version of Featherweight Java.
|
||||
Methods are declared without parameter or return types.
|
||||
We still keep type annotations for fields and generic class parameters.
|
||||
This is a design choice by us,
|
||||
as we consider them as data declarations which are given by the programmer.
|
||||
% They are inferred in for example \cite{plue14_3b}
|
||||
Note the \texttt{new} expression not requiring generic parameters,
|
||||
which are also inferred by our algorithm.
|
||||
The method call naturally has no generic parameters aswell.
|
||||
We add the elvis operator ($\elvis{}$) to the syntax mainly to showcase applications involving wildcard types.
|
||||
The syntax is described in figure \ref{fig:inputSyntax}.
|
||||
|
||||
|
||||
\begin{figure}
|
||||
$
|
||||
\begin{array}{lrcl}
|
||||
\hline
|
||||
\text{Parameterized classes} & \mv N & ::= & \exptype{C}{\ol{T}} \\
|
||||
\text{Types} & \type{S}, \type{T}, \type{U} & ::= & \type{X} \mid \wcNtype{\Delta}{N} \\
|
||||
\text{Lower bounds} & \type{K}, \type{L} & ::= & \type{T} \mid \bot \\
|
||||
\text{Type variable contexts} & \Delta & ::= & \overline{\wildcard{X}{T}{L}} \\
|
||||
\text{Class declarations} & D & ::= & \texttt{class}\ \exptype{C}{\ol{X \triangleleft T}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M}} \\
|
||||
\text{Method declarations} & \texttt{M} & ::= & \texttt{m}(\overline{x}) \{ \texttt{return}\ t; \} \\
|
||||
\text{Terms} & t & ::= & x \\
|
||||
& & \ \ | & \texttt{new} \ \type{C}(\overline{t})\\
|
||||
& & \ \ | & t.f\\
|
||||
& & \ \ | & t.\texttt{m}(\overline{t})\\
|
||||
& & \ \ | & t \elvis{} t\\
|
||||
\text{Variable contexts} & \Gamma & ::= & \overline{x:\type{T}}\\
|
||||
\hline
|
||||
\end{array}
|
||||
$
|
||||
\caption{Input Syntax}\label{fig:inputSyntax}
|
||||
\end{figure}
|
||||
|
||||
The output is a valid Featherweight Java program.
|
||||
We use the syntax of the version introduced in \cite{WildcardsNeedWitnessProtection}
|
||||
calling it \letfj{} for that it is a Featherweight Java variant including \texttt{let} statements.
|
||||
Our output syntax is shown in figure \ref{fig:outputSyntax}
|
||||
which is actually a subset of \letfj{}, because we omit \texttt{null} types.
|
||||
|
||||
\begin{figure}
|
||||
$
|
||||
\begin{array}{lrcl}
|
||||
\hline
|
||||
\text{Parameterized classes} & \mv N & ::= & \exptype{C}{\ol{T}} \\
|
||||
\text{Types} & \type{S}, \type{T}, \type{U} & ::= & \type{X} \mid \wcNtype{\Delta}{N} \\
|
||||
\text{Lower bounds} & \type{K}, \type{L} & ::= & \type{T} \mid \bot \\
|
||||
\text{Type variable contexts} & \Delta & ::= & \overline{\wildcard{X}{T}{L}} \\
|
||||
\text{Class declarations} & D & ::= & \texttt{class}\ \exptype{C}{\ol{X \triangleleft T}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M}} \\
|
||||
\text{Method declarations} & \texttt{M} & ::= & \generics{\overline{\type{X} \triangleleft \type{N}}}\type{T}\ \texttt{m}(\overline{\type{T}\ \expr{x}}) = \expr{e} \\
|
||||
\text{Terms} & \expr{e} & ::= & \expr{x} \\
|
||||
& & \ \ | & \texttt{new} \ \exptype{C}{\ol{T}}(\overline{\expr{e}})\\
|
||||
& & \ \ | & \expr{e}.f\\
|
||||
& & \ \ | & \expr{e}.\texttt{m}\generics{\ol{T}}(\overline{\expr{e}})\\
|
||||
& & \ \ | & \texttt{let}\ \expr{x} : \wcNtype{\Delta}{N} = \expr{e} \ \texttt{in} \ \expr{e}\\
|
||||
& & \ \ | & \expr{e} \elvis{} \expr{e}\\
|
||||
\text{Variable contexts} & \Gamma & ::= & \overline{\expr{x}:\type{T}}\\
|
||||
\hline
|
||||
\end{array}
|
||||
$
|
||||
\caption{Output Syntax}\label{fig:outputSyntax}
|
||||
\end{figure}
|
||||
|
||||
|
||||
The output of our type inference algorithm is a valid \letfj{} program.
|
||||
Before generating constraints the input is transformed to \TamedFJ{}.
|
||||
After adding the missing type annotations the resulting program is a valid \letfj{} program.
|
||||
%This is shown in chapter \ref{sec:soundness}
|
||||
Capture conversion is only needed for wildcard types,
|
||||
but we don't know which expressions will spawn wildcard types because there are no type annotations yet.
|
||||
We preemptively enclose every expression in a let statement before using it as a method argument.
|
||||
|
||||
We need the let statements to deal with possible Wildcard types.
|
||||
|
||||
|
||||
The syntax used in our examples is the standard Featherweight Java syntax.
|
||||
|
||||
|
||||
\subsection{Challenges}\label{challenges}
|
||||
%TODO: Wildcard subtyping is infinite see \cite{TamingWildcards}
|
||||
@ -414,7 +499,7 @@ it is safe to pass \texttt{"String"} for the first parameter of the function.
|
||||
<A> List<A> add(List<A> l, A v) {}
|
||||
|
||||
List<? super String> list = ...;
|
||||
add("String", list);
|
||||
add(list, "String");
|
||||
\end{verbatim}
|
||||
\end{example}
|
||||
|
||||
@ -460,7 +545,7 @@ class SpecialPair<X, Y extends X> extends Pair<X,Y>{}
|
||||
|
||||
|
||||
<X,Y> Pair<X,Y> id(Pair<X,Y> in){
|
||||
return ...;
|
||||
return in;
|
||||
}
|
||||
|
||||
<X, Y extends X> void receive(Pair<X,Y> in){}
|
||||
@ -531,154 +616,121 @@ $
|
||||
The \unify{} algorithm only sees the constraints with no information about the program they originated from.
|
||||
The main challenge was to find an algorithm which computes $\sigma(\wtv{a}) = \rwildcard{X}$ for example \ref{intro-example1} but not for example \ref{intro-example2}.
|
||||
|
||||
\section{\TamedFJ{} and \letfj{}}
|
||||
The input to our type inference algorithm is a \TamedFJ{} program.
|
||||
The output is a valid \letfj{} program.
|
||||
\letfj{} is defined in \cite{WildcardsNeedWitnessProtection}.
|
||||
It is an extension to Featherweight Java which adds Wildcard types and let statements.
|
||||
The let statements are used to explicitly apply capture conversion.
|
||||
|
||||
%TODO
|
||||
% The goal is to proof soundness in respect to the type rules introduced by \cite{aModelForJavaWithWildcards}
|
||||
% and \cite{WildcardsNeedWitnessProtection}.
|
||||
|
||||
\subsection{Capture Conversion}
|
||||
The input to our type inference algorithm does not contain let statements.
|
||||
Those are added after computing a type solution.
|
||||
Let statements act as capture conversion and only have to be applied in method calls involving wildcard types.
|
||||
% \subsection{Capture Conversion}
|
||||
% The \texttt{let} statements in \TamedFJ{} act as capture conversion for wildcard types.
|
||||
|
||||
\begin{figure}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{lstlisting}[style=fgj]
|
||||
<X> List<X> clone(List<X> l);
|
||||
example(p){
|
||||
return clone(p);
|
||||
}
|
||||
\end{lstlisting}
|
||||
\end{minipage}%
|
||||
\hfill
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{lstlisting}[style=tfgj]
|
||||
(*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) example((*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) p){
|
||||
return let x : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) = p in
|
||||
clone(x) : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*);
|
||||
}
|
||||
\end{lstlisting}
|
||||
\end{minipage}
|
||||
\caption{Type inference adding capture conversion}\label{fig:addingLetExample}
|
||||
\end{figure}
|
||||
% \begin{figure}
|
||||
% \begin{minipage}{0.45\textwidth}
|
||||
% \begin{lstlisting}[style=tfgj]
|
||||
% <X> List<X> clone(List<X> l);
|
||||
% example(p){
|
||||
% return clone(p);
|
||||
% }
|
||||
% \end{lstlisting}
|
||||
% \end{minipage}%
|
||||
% \hfill
|
||||
% \begin{minipage}{0.5\textwidth}
|
||||
% \begin{lstlisting}[style=letfj]
|
||||
% <X> List<X> clone(List<X> l);
|
||||
% (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) example((*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) p) =
|
||||
% let x : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) = p in
|
||||
% clone(x) : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*);
|
||||
% \end{lstlisting}
|
||||
% \end{minipage}
|
||||
% \caption{Type inference adding capture conversion}\label{fig:addingLetExample}
|
||||
% \end{figure}
|
||||
|
||||
Figure \ref{fig:addingLetExample} shows a let statement getting added to the typed output.
|
||||
The method \texttt{clone} cannot be called with the type $\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$.
|
||||
After a capture conversion \texttt{x} has the type $\exptype{List}{\rwildcard{X}}$ with $\rwildcard{X}$ being a free variable.
|
||||
Afterwards we have to find a supertype of $\exptype{List}{\rwildcard{X}}$, which does not contain free variables
|
||||
($\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ in this case).
|
||||
% Figure \ref{fig:addingLetExample} shows a let statement getting added to the typed output.
|
||||
% The method \texttt{clone} cannot be called with the type $\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$.
|
||||
% After a capture conversion \texttt{x} has the type $\exptype{List}{\rwildcard{X}}$ with $\rwildcard{X}$ being a free variable.
|
||||
% Afterwards we have to find a supertype of $\exptype{List}{\rwildcard{X}}$, which does not contain free variables
|
||||
% ($\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ in this case).
|
||||
|
||||
During the constraint generation step most types are not known yet and are represented by a type placeholder.
|
||||
During a methodcall like the one in the \texttt{example} method in figure \ref{fig:ccExample} the type of the parameter \texttt{p}
|
||||
is not known yet.
|
||||
The type \texttt{List<?>} would be one possibility as a parameter type for \texttt{p}.
|
||||
To make wildcards work for our type inference algorithm \unify{} has to apply capture conversions if necessary.
|
||||
% During the constraint generation step most types are not known yet and are represented by a type placeholder.
|
||||
% During a methodcall like the one in the \texttt{example} method in figure \ref{fig:ccExample} the type of the parameter \texttt{p}
|
||||
% is not known yet.
|
||||
% The type \texttt{List<?>} would be one possibility as a parameter type for \texttt{p}.
|
||||
% To make wildcards work for our type inference algorithm \unify{} has to apply capture conversions if necessary.
|
||||
|
||||
The type placeholder $\tv{r}$ is the return type of the \texttt{example} method.
|
||||
One possible type solution is $\tv{p} \doteq \tv{r} \doteq \wctype{\rwildcard{X}}{List}{\rwildcard{X}}$,
|
||||
which leads to:
|
||||
\begin{verbatim}
|
||||
List<?> example(List<?> p){
|
||||
return clone(p);
|
||||
}
|
||||
\end{verbatim}
|
||||
% The type placeholder $\tv{r}$ is the return type of the \texttt{example} method.
|
||||
% One possible type solution is $\tv{p} \doteq \tv{r} \doteq \wctype{\rwildcard{X}}{List}{\rwildcard{X}}$,
|
||||
% which leads to:
|
||||
% \begin{verbatim}
|
||||
% List<?> example(List<?> p){
|
||||
% return clone(p);
|
||||
% }
|
||||
% \end{verbatim}
|
||||
|
||||
But by substituting $\tv{p} \doteq \wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ in the constraint
|
||||
$\tv{p} \lessdotCC \exptype{List}{\wtv{x}}$ leads to
|
||||
$\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{x}}$.
|
||||
% But by substituting $\tv{p} \doteq \wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ in the constraint
|
||||
% $\tv{p} \lessdotCC \exptype{List}{\wtv{x}}$ leads to
|
||||
% $\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{x}}$.
|
||||
|
||||
To make this typing possible we have to introduce a capture conversion via a let statement:
|
||||
$\texttt{return}\ (\texttt{let}\ \texttt{x} : \wctype{\rwildcard{X}}{List}{\rwildcard{X}} = \texttt{p}\ \texttt{in} \
|
||||
\texttt{clone}\generics{\rwildcard{X}}(x) : \wctype{\rwildcard{X}}{List}{\rwildcard{X}})$
|
||||
% To make this typing possible we have to introduce a capture conversion via a let statement:
|
||||
% $\texttt{return}\ (\texttt{let}\ \texttt{x} : \wctype{\rwildcard{X}}{List}{\rwildcard{X}} = \texttt{p}\ \texttt{in} \
|
||||
% \texttt{clone}\generics{\rwildcard{X}}(x) : \wctype{\rwildcard{X}}{List}{\rwildcard{X}})$
|
||||
|
||||
Inside the let statement the variable \texttt{x} has the type $\exptype{List}{\rwildcard{X}}$
|
||||
% Inside the let statement the variable \texttt{x} has the type $\exptype{List}{\rwildcard{X}}$
|
||||
|
||||
|
||||
This spawns additional problems.
|
||||
% This spawns additional problems.
|
||||
|
||||
\begin{figure}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{verbatim}
|
||||
<X> List<X> clone(List<X> l){...}
|
||||
% \begin{figure}
|
||||
% \begin{minipage}{0.45\textwidth}
|
||||
% \begin{verbatim}
|
||||
% <X> List<X> clone(List<X> l){...}
|
||||
|
||||
example(p){
|
||||
return clone(p);
|
||||
}
|
||||
\end{verbatim}
|
||||
\end{minipage}%
|
||||
\hfill
|
||||
\begin{minipage}{0.35\textwidth}
|
||||
\begin{constraintset}
|
||||
\textbf{Constraints:}\\
|
||||
$
|
||||
\tv{p} \lessdotCC \exptype{List}{\wtv{x}}, \\
|
||||
\tv{p} \lessdot \tv{r}, \\
|
||||
\tv{p} \lessdot \type{Object},
|
||||
\tv{r} \lessdot \type{Object}
|
||||
$
|
||||
\end{constraintset}
|
||||
\end{minipage}
|
||||
% example(p){
|
||||
% return clone(p);
|
||||
% }
|
||||
% \end{verbatim}
|
||||
% \end{minipage}%
|
||||
% \hfill
|
||||
% \begin{minipage}{0.35\textwidth}
|
||||
% \begin{constraintset}
|
||||
% \textbf{Constraints:}\\
|
||||
% $
|
||||
% \tv{p} \lessdotCC \exptype{List}{\wtv{x}}, \\
|
||||
% \tv{p} \lessdot \tv{r}, \\
|
||||
% \tv{p} \lessdot \type{Object},
|
||||
% \tv{r} \lessdot \type{Object}
|
||||
% $
|
||||
% \end{constraintset}
|
||||
% \end{minipage}
|
||||
|
||||
\caption{Type inference example}\label{fig:ccExample}
|
||||
\end{figure}
|
||||
% \caption{Type inference example}\label{fig:ccExample}
|
||||
% \end{figure}
|
||||
|
||||
In addition with free variables this leads to unwanted behaviour.
|
||||
Take the constraint
|
||||
$\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$ for example.
|
||||
After a capture conversion from $\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ to $\exptype{List}{\rwildcard{Y}}$ and a substitution $\wtv{a} \doteq \rwildcard{Y}$
|
||||
we get
|
||||
$\exptype{List}{\rwildcard{Y}} \lessdot \exptype{List}{\rwildcard{Y}}$.
|
||||
Which is correct if we apply capture conversion to the left side:
|
||||
$\exptype{List}{\rwildcard{X}} <: \exptype{List}{\rwildcard{X}}$
|
||||
If the input constraints did not intend for this constraint to undergo a capture conversion then \unify{} would produce an invalid
|
||||
type solution due to:
|
||||
$\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \nless: \exptype{List}{\rwildcard{X}}$
|
||||
The reason for this is the \texttt{S-Exists} rule's premise
|
||||
$\text{dom}(\Delta') \cap \text{fv}(\exptype{List}{\rwildcard{X}}) = \emptyset$.
|
||||
% In addition with free variables this leads to unwanted behaviour.
|
||||
% Take the constraint
|
||||
% $\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$ for example.
|
||||
% After a capture conversion from $\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ to $\exptype{List}{\rwildcard{Y}}$ and a substitution $\wtv{a} \doteq \rwildcard{Y}$
|
||||
% we get
|
||||
% $\exptype{List}{\rwildcard{Y}} \lessdot \exptype{List}{\rwildcard{Y}}$.
|
||||
% Which is correct if we apply capture conversion to the left side:
|
||||
% $\exptype{List}{\rwildcard{X}} <: \exptype{List}{\rwildcard{X}}$
|
||||
% If the input constraints did not intend for this constraint to undergo a capture conversion then \unify{} would produce an invalid
|
||||
% type solution due to:
|
||||
% $\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \nless: \exptype{List}{\rwildcard{X}}$
|
||||
% The reason for this is the \texttt{S-Exists} rule's premise
|
||||
% $\text{dom}(\Delta') \cap \text{fv}(\exptype{List}{\rwildcard{X}}) = \emptyset$.
|
||||
|
||||
Additionally free variables are not allowed to leave the scope of a capture conversion
|
||||
introduced by a let statement.
|
||||
%TODO we combat both of this with wildcard type placeholders (? flag)
|
||||
|
||||
Type placeholders which are not flagged as possible free variables ($\wtv{a}$) can never hold a free variable or a type containing free variables.
|
||||
Constraint generation places these standard place holders at method return types and parameter types.
|
||||
\begin{lstlisting}[style=fgj]
|
||||
<X> List<X> clone(List<X> l);
|
||||
(*@$\red{\tv{r}}$@*) example((*@$\red{\tv{p}}$@*) p){
|
||||
return clone(p);
|
||||
}
|
||||
\end{lstlisting}
|
||||
This prevents type solutions that contain free variables in parameter and return types.
|
||||
When calling a method which already has a type annotation we have to consider adding a capture conversion in form of a let statement.
|
||||
The constraint $\tv{p} \lessdot \exptype{List}{\wtv{x}}$ signals the \unify{} algorithm that here a capture conversion is possible.
|
||||
$\sigma(\tv{p}) = \wctype{\rwildcard{X}}{List}{\rwildcard{X}}, \sigma(\tv{r}) = \wctype{\rwildcard{X}}{List}{\rwildcard{X}}, $ is a possible solution.
|
||||
But only when adding a capture conversion:
|
||||
\begin{lstlisting}[style=fgj]
|
||||
<X> List<X> clone(List<X> l);
|
||||
(*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) example((*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) p){
|
||||
return let x : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) = p in clone(x) : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*);
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
||||
Capture constraints cannot be stored in a set.
|
||||
$\wtv{a} \lessdotCC \wtv{b}$ is not the same as $\wtv{a} \lessdotCC \wtv{b}$.
|
||||
Both constraints will end up the same after a substitution for both placeholders $\tv{a}$ and $\tv{b}$.
|
||||
But afterwards a capture conversion is applied, which can generate different types on the left sides.
|
||||
\begin{itemize}
|
||||
\item $\text{CC}(\wctype{\rwildcard{X}}{List}{\rwildcard{X}}) \implies \exptype{List}{\rwildcard{Y}}$
|
||||
\item $\text{CC}(\wctype{\rwildcard{X}}{List}{\rwildcard{X}}) \implies \exptype{List}{\rwildcard{Z}}$
|
||||
\end{itemize}
|
||||
|
||||
Wildcards are not reflexive. A box of type $\wctype{\rwildcard{X}}{Box}{\rwildcard{X}}$
|
||||
is able to hold a value of any type. It could be a $\exptype{Box}{String}$ or a $\exptype{Box}{Integer}$ etc.
|
||||
Also two of those boxes do not necessarily contain the same type.
|
||||
But there are situations where it is possible to assume that.
|
||||
For example the type $\wctype{\rwildcard{X}}{Pair}{\exptype{Box}{\rwildcard{X}}, \exptype{Box}{\rwildcard{X}}}$
|
||||
is a pair of two boxes, which always contain the same type.
|
||||
Inside the scope of the \texttt{Pair} type, the wildcard $\rwildcard{X}$ stays the same.
|
||||
% Capture constraints cannot be stored in a set.
|
||||
% $\wtv{a} \lessdotCC \wtv{b}$ is not the same as $\wtv{a} \lessdotCC \wtv{b}$.
|
||||
% Both constraints will end up the same after a substitution for both placeholders $\tv{a}$ and $\tv{b}$.
|
||||
% But afterwards a capture conversion is applied, which can generate different types on the left sides.
|
||||
% \begin{itemize}
|
||||
% \item $\text{CC}(\wctype{\rwildcard{X}}{List}{\rwildcard{X}}) \implies \exptype{List}{\rwildcard{Y}}$
|
||||
% \item $\text{CC}(\wctype{\rwildcard{X}}{List}{\rwildcard{X}}) \implies \exptype{List}{\rwildcard{Z}}$
|
||||
% \end{itemize}
|
||||
%
|
||||
% Wildcards are not reflexive. A box of type $\wctype{\rwildcard{X}}{Box}{\rwildcard{X}}$
|
||||
% is able to hold a value of any type. It could be a $\exptype{Box}{String}$ or a $\exptype{Box}{Integer}$ etc.
|
||||
% Also two of those boxes do not necessarily contain the same type.
|
||||
% But there are situations where it is possible to assume that.
|
||||
% For example the type $\wctype{\rwildcard{X}}{Pair}{\exptype{Box}{\rwildcard{X}}, \exptype{Box}{\rwildcard{X}}}$
|
||||
% is a pair of two boxes, which always contain the same type.
|
||||
% Inside the scope of the \texttt{Pair} type, the wildcard $\rwildcard{X}$ stays the same.
|
||||
@ -13,7 +13,8 @@
|
||||
}
|
||||
\lstdefinestyle{fgj}{backgroundcolor=\color{lime!20}}
|
||||
\lstdefinestyle{tfgj}{backgroundcolor=\color{lightgray!20}}
|
||||
\lstdefinestyle{letfj}{backgroundcolor=\color{lightgray!20}}
|
||||
\lstdefinestyle{letfj}{backgroundcolor=\color{lime!20}}
|
||||
\lstdefinestyle{tamedfj}{backgroundcolor=\color{yellow!20}}
|
||||
\lstdefinestyle{java}{backgroundcolor=\color{lightgray!20}}
|
||||
|
||||
\newtheorem{recap}[note]{Recap}
|
||||
|
||||
72
tRules.tex
72
tRules.tex
@ -61,7 +61,7 @@ $
|
||||
\hline
|
||||
\end{array}
|
||||
$
|
||||
\caption{Input Syntax}\label{fig:syntax}
|
||||
\caption{\TamedFJ{} Syntax}\label{fig:syntax}
|
||||
\end{figure}
|
||||
|
||||
% \begin{figure}
|
||||
@ -139,15 +139,16 @@ Featherweight Java's syntax involves no \texttt{let} statement
|
||||
and terms can be nested freely.
|
||||
This is similar to Java's syntax.
|
||||
To convert it to \TamedFJ{} additional let statements have to be added.
|
||||
This is done by a \textit{A-Normal Form} \cite{aNormalForm} transformation.
|
||||
|
||||
This is done by a \textit{A-Normal Form} \cite{aNormalForm} transformation shown in figure \ref{fig:anfTransformation}.
|
||||
The input of this transformation is a Featherweight Java program in the syntax given \ref{fig:inputSyntax}
|
||||
and the output is a \TamedFJ{} program.
|
||||
|
||||
\textit{Example:}\\
|
||||
\noindent
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{lstlisting}[style=fgj,caption=Featherweight Java]
|
||||
m(l, v){
|
||||
return id(l).add(v);
|
||||
return l.add(v);
|
||||
}
|
||||
\end{lstlisting}
|
||||
\end{minipage}%
|
||||
@ -155,42 +156,48 @@ m(l, v){
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{lstlisting}[style=tfgj,caption=\TamedFJ{} representation]
|
||||
m(l, v) =
|
||||
let x = id(l) in x.add(v)
|
||||
|
||||
let x1 = l in
|
||||
let x2 = v in x1.add(x2)
|
||||
\end{lstlisting}
|
||||
\end{minipage}
|
||||
|
||||
|
||||
$
|
||||
\begin{array}{|lrcl|l}
|
||||
\hline
|
||||
& & & \textbf{Featherweight Java Terms}\\
|
||||
\text{Terms} & t & ::=
|
||||
& \expr{x}
|
||||
\\
|
||||
& & \ \ |
|
||||
& \texttt{new} \ \type{C}(\overline{t})
|
||||
\\
|
||||
& & \ \ |
|
||||
& t.f
|
||||
\\
|
||||
& & \ \ |
|
||||
& t.\texttt{m}(\overline{t})
|
||||
\\
|
||||
& & \ \ |
|
||||
& t \elvis{} t\\
|
||||
%
|
||||
\hline
|
||||
\end{array}
|
||||
$
|
||||
% $
|
||||
% \begin{array}{|lrcl|l}
|
||||
% \hline
|
||||
% & & & \textbf{Featherweight Java Terms}\\
|
||||
% \text{Terms} & t & ::=
|
||||
% & \expr{x}
|
||||
% \\
|
||||
% & & \ \ |
|
||||
% & \texttt{new} \ \type{C}(\overline{t})
|
||||
% \\
|
||||
% & & \ \ |
|
||||
% & t.f
|
||||
% \\
|
||||
% & & \ \ |
|
||||
% & t.\texttt{m}(\overline{t})
|
||||
% \\
|
||||
% & & \ \ |
|
||||
% & t \elvis{} t\\
|
||||
% %
|
||||
% \hline
|
||||
% \end{array}
|
||||
% $
|
||||
|
||||
\begin{figure}
|
||||
\begin{center}
|
||||
$\begin{array}{lrcl}
|
||||
\text{ANF} & \anf{\expr{x}} & = & \expr{x} \\
|
||||
%\text{ANF}
|
||||
& \anf{\expr{x}} & = & \expr{x} \\
|
||||
& \anf{\texttt{new} \ \type{C}(\overline{t})} & = & \texttt{let}\ \overline{\expr{x}} = \anf{\overline{t}} \ \texttt{in}\ \texttt{new} \ \type{C}(\overline{\expr{x}}) \\
|
||||
& \anf{t.f} & = & \texttt{let}\ \expr{x} = \anf{t} \ \texttt{in}\ \expr{x}.f \\
|
||||
& \anf{t.\texttt{m}(\overline{t})} & = & \texttt{let}\ \expr{x} = \anf{t} \ \texttt{in}\ \texttt{let}\ \overline{\expr{x}} = \anf{\overline{t}} \ \texttt{in}\ \expr{x}.\texttt{m}(\overline{\expr{x}}) \\
|
||||
& \anf{t_1 \elvis{} t_2} & = & \anf{t_1} \elvis{} \anf{t_2}
|
||||
\end{array}$
|
||||
\end{center}
|
||||
\caption{ANF Transformation}\label{fig:anfTransformation}
|
||||
\end{figure}
|
||||
|
||||
% $
|
||||
% \begin{array}{lrcl|l}
|
||||
@ -455,7 +462,7 @@ $\begin{array}{l}
|
||||
$\begin{array}{l}
|
||||
\typerule{T-Let}\\
|
||||
\begin{array}{@{}c}
|
||||
\Delta | \Gamma \vdash \expr{e}_1 : \type{T}_1 \quad \quad
|
||||
\Delta | \Gamma \vdash \expr{t}_1 : \type{T}_1 \quad \quad
|
||||
\Delta \vdash \type{T}_1 <: \wcNtype{\Delta'}{N}
|
||||
\\
|
||||
\Delta, \Delta' | \Gamma, \expr{x} : \wcNtype{}{N} \vdash \expr{t}_2 : \type{T}_2 \quad \quad
|
||||
@ -465,7 +472,7 @@ $\begin{array}{l}
|
||||
\\
|
||||
\hline
|
||||
\vspace*{-0.3cm}\\
|
||||
\Delta | \Gamma \vdash \texttt{let}\ \expr{x} = \expr{e}_1 \ \texttt{in} \ \expr{e}_2 : \type{T}
|
||||
\Delta | \Gamma \vdash \texttt{let}\ \expr{x} = \expr{t}_1 \ \texttt{in} \ \expr{t}_2 : \type{T}
|
||||
\end{array}
|
||||
\end{array}$
|
||||
\\[1em]
|
||||
@ -508,8 +515,7 @@ $\begin{array}{l}
|
||||
$\begin{array}{l}
|
||||
\typerule{T-Class}\\
|
||||
\begin{array}{@{}c}
|
||||
\mathtt{\Pi}' = \mathtt{\Pi} \cup \set{ \texttt{m} : (\type{T'_m}, \ol{T_\texttt{m}}) \to \type{T}_\texttt{m} \mid \texttt{m} \in \ol{M}} \quad \quad
|
||||
\forall \texttt{m} \in \ol{M}: \Delta \vdash \exptype{C}{\ol{X}} <: \type{T'_m}\\
|
||||
\mathtt{\Pi}' = \mathtt{\Pi} \cup \set{ \texttt{m} : (\exptype{C}{\ol{X}}, \ol{T_\texttt{m}}) \to \type{T}_\texttt{m} \mid \texttt{m} \in \ol{M}} \\
|
||||
\mathtt{\Pi}'' = \mathtt{\Pi} \cup \set{ \texttt{m} :
|
||||
\generics{\ol{X \triangleleft \type{N}}, \ol{Y \triangleleft P}}(\exptype{C}{\ol{X}},\ol{T_\texttt{m}}) \to \type{T}_\texttt{m} \mid \texttt{m} \in \ol{M} } \\
|
||||
\triangle = \overline{\type{X} : \bot .. \type{U}} \quad \quad
|
||||
|
||||
Loading…
Reference in New Issue
Block a user