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Andreas Stadelmeier 2024-05-31 00:10:22 +02:00
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37
TIforWildFJ.tex
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@ -112,51 +112,26 @@
\input{tRules} \input{tRules}
\input{typeinference}
%\input{tiRules} %\input{tiRules}
\input{constraints} \input{constraints}
\input{Unify} \input{Unify}
\section{Limitations} \include{relatedwork}
This example does not work:
\begin{minipage}{0.35\textwidth}
\begin{verbatim}
class Example{
<A> Pair<A,A> make(List<A> l){...}
<A> bool compare(Pair<A,A> p){...}
bool test(List<?> l){
return compare(make(l));
}
}
\end{verbatim}
\end{minipage}%
\hfill
\begin{minipage}{0.55\textwidth}
\begin{constraintset}
\textbf{Constraints:}\\
$
\wctype{\wildcard{A}{\type{Object}}{\bot}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\wtv{x}}, \\
\exptype{Pair}{\wtv{x},\wtv{x}} \lessdot \tv{r}, \\
\tv{r} \lessdot \exptype{Pair}{\tv{z}, \tv{z}}%,\\
%\tv{y} \lessdot \tv{m}
$\\
\end{constraintset}
\end{minipage}
\input{conclusion} \input{conclusion}
\include{soundness}
%\include{termination} %\include{termination}
\include{relatedwork}
\bibliography{martin} \bibliography{martin}
\appendix \appendix
\include{soundness}
\include{helpers}
%\include{examples} %\include{examples}
%\input{exampleWildcardParameter} %\input{exampleWildcardParameter}
%\input{commonPatternsProof} %\input{commonPatternsProof}

49
conclusion.tex
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@ -9,7 +9,56 @@
%TODO: how are the challenges solved: Describe this in the last chapter with examples! %TODO: how are the challenges solved: Describe this in the last chapter with examples!
\section{Soundness}\label{sec:soundness} \section{Soundness}\label{sec:soundness}
\begin{theorem}\label{testenv-theorem}
Type inference produces a correctly typed program.
\begin{description}
\item[If] $\fjtypeinference(\mv{\Pi}, \texttt{class}\ \exptype{C}{\ol{X}
\triangleleft \ol{N}} \triangleleft \type{N}\ \{ \overline{\type{T} \ f};\ \ol{M} \}) = \mtypeEnvironment{}'$
\item[Then] $\texttt{class}\ \exptype{C}{\ol{X}
\triangleleft \ol{N}} \triangleleft \type{N}\ \{ \overline{\type{T} \ f};\ \ol{M} \} \text{ok}$,
with $\ol{M} = $
\end{description}
\end{theorem}
To prove soundness we have to show two things.
The vital parts are method calls and field access.
The generated constraints have to resemble the \TamedFJ{}'s type system.
\section{Completeness}\label{sec:completeness} \section{Completeness}\label{sec:completeness}
We couldn't verify it with an implementation yet, but we assume atleast the same functionality
as the global type inference algorithm for Featherweight Java without Wildcards \cite{TIforFGJ}.
\begin{example} Limitation:
This example does not work:
\begin{minipage}{0.35\textwidth}
\begin{verbatim}
class Example{
<A> Pair<A,A> make(List<A> l){...}
<A> bool compare(Pair<A,A> p){...}
bool test(List<?> l){
return compare(make(l));
}
}
\end{verbatim}
\end{minipage}%
\hfill
\begin{minipage}{0.55\textwidth}
\begin{constraintset}
\textbf{Constraints:}\\
$
\wctype{\wildcard{A}{\type{Object}}{\bot}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\wtv{x}}, \\
\exptype{Pair}{\wtv{x},\wtv{x}} \lessdot \tv{r}, \\
\tv{r} \lessdot \exptype{Pair}{\tv{z}, \tv{z}}%,\\
%\tv{y} \lessdot \tv{m}
$\\
\end{constraintset}
\end{minipage}
\end{example}
The algorithm can find a solution to every program which the Unify by Plümicke finds The algorithm can find a solution to every program which the Unify by Plümicke finds
a correct solution aswell. a correct solution aswell.
It will not infer intermediat type like $\wctype{\rwildcard{X}}{Pair}{\rwildcard{X},\rwildcard{X}}$. It will not infer intermediat type like $\wctype{\rwildcard{X}}{Pair}{\rwildcard{X},\rwildcard{X}}$.

102
constraints.tex
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@ -1,70 +1,3 @@
\subsection{Capture Constraints}
%TODO: General Capture Constraint explanation
Capture Constraints are bound to a variable.
For example a let statement like \lstinline{let x = v in x.get()} will create the capture constraint
$\tv{x} \lessdotCC_x \exptype{List}{\wtv{a}}$.
This time we annotated the capture constraint with an $x$ to show its relation to the variable \texttt{x}.
Let's do the same with the constraints generated by the \texttt{concat} method invocation in listing \ref{lst:faultyConcat},
creating the constraints \ref{lst:sameConstraints}.
\begin{figure}
\begin{minipage}[t]{0.49\textwidth}
\begin{lstlisting}[caption=Faulty Method Call,label=lst:faultyConcat]{tamedfj}
List<?> v = ...;
let x = v in
let y = v in
concat(x, y) // Error!
\end{lstlisting}\end{minipage}
\hfill
\begin{minipage}[t]{0.49\textwidth}
\begin{lstlisting}[caption=Annotated constraints,mathescape=true,style=constraints,label=lst:sameConstraints]
$\tv{x} \lessdotCC_\texttt{x} \exptype{List}{\wtv{a}}, \wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdot \tv{x}$
$\tv{y} \lessdotCC_\texttt{y} \exptype{List}{\wtv{a}}, \wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdot \tv{y}$
\end{lstlisting}
\end{minipage}
\end{figure}
During the \unify{} process it could happen that two syntactically equal capture constraints evolve,
but they are not the same because they are each linked to a different let introduced variable.
In this example this happens when we substitute $\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ for $\tv{x}$ and $\tv{y}$
resulting in:
%For example by substituting $[\wctype{\rwildcard{X}}{List}{\rwildcard{X}}/\tv{x}]$ and $[\wctype{\rwildcard{X}}{List}{\rwildcard{X}}/\tv{y}]$:
\begin{displaymath}
\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC_x \exptype{List}{\wtv{a}}, \wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC_y \exptype{List}{\wtv{a}}
\end{displaymath}
Thanks to the original annotations we can still see that those are different constraints.
After \unify{} uses the \rulename{Capture} rule on those constraints
it gets obvious that this constraint set is unsolvable:
\begin{displaymath}
\exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}},
\exptype{List}{\rwildcard{Y}} \lessdot \exptype{List}{\wtv{a}}
\end{displaymath}
%In this paper we do not annotate capture constraint with their source let statement.
The rest of this paper will not annotate capture constraints with variable names.
Instead we consider every capture constraint as distinct to other capture constraints even when syntactically the same,
because we know that each of them originates from a different let statement.
\textit{Hint:} An implementation of this algorithm has to consider that seemingly equal capture constraints are actually not the same
and has to allow doubles in the constraint set.
% %We see the equality relation on Capture constraints is not reflexive.
% A capture constraint is never equal to another capture constraint even when structurally the same
% ($\type{T} \lessdotCC \type{S} \neq \type{T} \lessdotCC \type{S}$).
% This is necessary to solve challenge \ref{challenge:1}.
% A capture constraint is bound to a specific let statement.
\textit{Note:}
In the special case \lstinline{let x = v in concat(x,x)} the constraints would look like
$\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC_\texttt{x} \exptype{List}{\wtv{a}},
\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC_\texttt{x} \exptype{List}{\wtv{a}}$
and we could actually delete one of them without loosing information.
But this case will never occur in our algorithm, because the let
statements for our input programs are generated by transformation to ANF (see \ref{sec:anfTransformation}).
\section{Constraint generation}\label{chapter:constraintGeneration} \section{Constraint generation}\label{chapter:constraintGeneration}
% Method names are not unique. % Method names are not unique.
@ -78,11 +11,6 @@ statements for our input programs are generated by transformation to ANF (see \r
% But it can be easily adapted to Featherweight Java or Java. % But it can be easily adapted to Featherweight Java or Java.
% 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 % 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
\begin{description}
\item[input] \TamedFJ{} program in A-Normal form
\item[output] Constraints
\end{description}
The constraint generation works on the \TamedFJ{} language. The constraint generation works on the \TamedFJ{} language.
This step is mostly the same as in \cite{TIforFGJ} except for field access and method invocation. This step is mostly the same as in \cite{TIforFGJ} except for field access and method invocation.
We will focus on those two parts where also the new capture constraints and wildcard type placeholders are introduced. We will focus on those two parts where also the new capture constraints and wildcard type placeholders are introduced.
@ -146,7 +74,7 @@ $\begin{array}{lrcl}
& \anf{\texttt{let}\ x = \expr{t}_1 \ \texttt{in}\ \expr{t}_2} & = & \texttt{let}\ x = \anf{\expr{t}_1} \ \texttt{in}\ \anf{\expr{t}_2} & \anf{\texttt{let}\ x = \expr{t}_1 \ \texttt{in}\ \expr{t}_2} & = & \texttt{let}\ x = \anf{\expr{t}_1} \ \texttt{in}\ \anf{\expr{t}_2}
\end{array}$ \end{array}$
\end{center} \end{center}
\caption{ANF Transformation}\label{fig:anfTransformation} \caption{ANF Transformation $\tau$}\label{fig:anfTransformation}
\end{figure} \end{figure}
\begin{figure} \begin{figure}
@ -216,7 +144,7 @@ Those type variables count as regular types and can be held by normal type place
\begin{figure}[tp] \begin{figure}[tp]
\begin{gather*} \begin{gather*}
\begin{array}{@{}l@{}l} \begin{array}{@{}l@{}l}
\fjtype & ({\mtypeEnvironment}, \mathtt{class } \ \exptype{C}{\overline{\type{X} \triangleleft \type{N}}} \ \mathtt{ extends } \ \mathtt{N \{ \overline{T} \ \overline{f}; \, \overline{M} \}}) =\\ \fjtype & ({\mtypeEnvironment }, \mathtt{class } \ \exptype{C}{\overline{\type{X} \triangleleft \type{N}}} \ \mathtt{ extends } \ \mathtt{N \{ \overline{T} \ \overline{f}; \, \overline{M} \}}) =\\
& \begin{array}{ll@{}l} & \begin{array}{ll@{}l}
\textbf{let} & \ol{\methodAssumption} = \textbf{let} & \ol{\methodAssumption} =
\set{ \mv{m} : (\exptype{C}{\ol{X}}, \ol{\tv{a}} \to \tv{a}) \mid \set{ \mv{m} : (\exptype{C}{\ol{X}}, \ol{\tv{a}} \to \tv{a}) \mid
@ -238,11 +166,11 @@ Those type variables count as regular types and can be held by normal type place
\begin{displaymath} \begin{displaymath}
\begin{array}{@{}l@{}l} \begin{array}{@{}l@{}l}
\typeExpr{} &({\mtypeEnvironment} , \texttt{e}.\texttt{f}, \tv{a}) = \\ \typeExpr{} &({\mtypeEnvironment}, \Gamma , \texttt{e}.\texttt{f}, \tv{a}) = \\
& \begin{array}{ll} & \begin{array}{ll}
\textbf{let} \textbf{let}
& \tv{r} \ \text{fresh} \\ & \tv{r} \ \text{fresh} \\
& \consSet_R = \typeExpr({\mtypeEnvironment}, \texttt{e}, \tv{r})\\ & \consSet_R = \typeExpr({\mtypeEnvironment}, \Gamma, \texttt{e}, \tv{r})\\
& \constraint = \begin{array}[t]{@{}l@{}l} & \constraint = \begin{array}[t]{@{}l@{}l}
\orCons\set{ \orCons\set{
\set{ & \set{ &
@ -261,12 +189,12 @@ Those type variables count as regular types and can be held by normal type place
\begin{displaymath} \begin{displaymath}
\begin{array}{@{}l@{}l} \begin{array}{@{}l@{}l}
\typeExpr{} &({\mtypeEnvironment} , \texttt{let}\ \expr{x} = \expr{e}_1 \ \texttt{in} \ \expr{e}_2, \tv{a}) = \\ \typeExpr{} &({\mtypeEnvironment}, \Gamma , \texttt{let}\ \expr{x} = \expr{e}_1 \ \texttt{in} \ \expr{e}_2, \tv{a}) = \\
& \begin{array}{ll} & \begin{array}{ll}
\textbf{let} \textbf{let}
& \tv{e}_1, \tv{e}_2, \tv{x} \ \text{fresh} \\ & \tv{e}_1, \tv{e}_2, \tv{x} \ \text{fresh} \\
& \consSet_1 = \typeExpr({\mtypeEnvironment}, \expr{e}_1, \tv{e}_1)\\ & \consSet_1 = \typeExpr({\mtypeEnvironment}, \Gamma, \expr{e}_1, \tv{e}_1)\\
& \consSet_2 = \typeExpr({\mtypeEnvironment} \cup \set{\expr{x} : \tv{x}}, \expr{e}_2, \tv{e}_2)\\ & \consSet_2 = \typeExpr({\mtypeEnvironment } \cup \set{\expr{x} : \tv{x}}, \expr{e}_2, \tv{e}_2)\\
& \constraint = & \constraint =
\set{ \set{
\tv{e}_1 \lessdot \tv{x}, \tv{e}_2 \lessdot \tv{a} \tv{e}_1 \lessdot \tv{x}, \tv{e}_2 \lessdot \tv{a}
@ -279,7 +207,7 @@ Those type variables count as regular types and can be held by normal type place
\begin{displaymath} \begin{displaymath}
\begin{array}{@{}l@{}l} \begin{array}{@{}l@{}l}
\typeExpr{} & ({\mtypeEnvironment} , \expr{v}.\mathtt{m}(\overline{\expr{v}}), \tv{a}) = \\ \typeExpr{} & ({\mtypeEnvironment}, \Gamma , \expr{v}.\mathtt{m}(\overline{\expr{v}}), \tv{a}) = \\
& \begin{array}{ll} & \begin{array}{ll}
\textbf{let} \textbf{let}
& \tv{r}, \ol{\tv{r}} \text{ fresh} \\ & \tv{r}, \ol{\tv{r}} \text{ fresh} \\
@ -289,7 +217,7 @@ Those type variables count as regular types and can be held by normal type place
\mathbf{in} & (\consSet_R \cup \overline{\consSet} \cup \constraint, \type{T}) \\ \mathbf{in} & (\consSet_R \cup \overline{\consSet} \cup \constraint, \type{T}) \\
& \mathbf{where}\ \begin{array}[t]{l} & \mathbf{where}\ \begin{array}[t]{l}
\expr{v}, \ol{v} : \ol{S} \in \localVarAssumption \\ \expr{v}, \ol{v} : \ol{S} \in \localVarAssumption \\
\texttt{m} : \generics{\ol{Y} \triangleleft \ol{N}}\overline{\type{T}} \to \type{T} \in {\mtypeEnvironment} \texttt{m} : \generics{\ol{Y} \triangleleft \ol{N}}\overline{\type{T}} \to \type{T} \in {\mtypeEnvironment }
\end{array} \end{array}
\end{array} \end{array}
@ -390,12 +318,12 @@ cannot be used anywhere else then inside the constraints generated by the method
\begin{displaymath} \begin{displaymath}
\begin{array}{@{}l@{}l} \begin{array}{@{}l@{}l}
\typeExpr{} &({\mtypeEnvironment} , e_1 \elvis{} e_2, \tv{a}) = \\ \typeExpr{} &({\mtypeEnvironment}, \Gamma , e_1 \elvis{} e_2, \tv{a}) = \\
& \begin{array}{ll} & \begin{array}{ll}
\textbf{let} \textbf{let}
& \tv{r}_1, \tv{r}_2 \ \text{fresh} \\ & \tv{r}_1, \tv{r}_2 \ \text{fresh} \\
& \consSet_1 = \typeExpr({\mtypeEnvironment}, e_1, \tv{r}_2)\\ & \consSet_1 = \typeExpr({\mtypeEnvironment}, \Gamma, e_1, \tv{r}_2)\\
& \consSet_2 = \typeExpr({\mtypeEnvironment}, e_2, \tv{r}_2)\\ & \consSet_2 = \typeExpr({\mtypeEnvironment}, \Gamma, e_2, \tv{r}_2)\\
{\mathbf{in}} & { {\mathbf{in}} & {
\consSet_1 \cup \consSet_2 \cup \consSet_1 \cup \consSet_2 \cup
\set{\tv{r}_1 \lessdot \tv{a}, \tv{r}_2 \lessdot \tv{a}}} \set{\tv{r}_1 \lessdot \tv{a}, \tv{r}_2 \lessdot \tv{a}}}
@ -406,18 +334,18 @@ cannot be used anywhere else then inside the constraints generated by the method
%We could skip wfresh here: %We could skip wfresh here:
\begin{displaymath} \begin{displaymath}
\begin{array}{@{}l@{}l} \begin{array}{@{}l@{}l}
\typeExpr{} &({\mtypeEnvironment} , x, \tv{a}) = \typeExpr{} &({\mtypeEnvironment}, \Gamma , x, \tv{a}) =
\mtypeEnvironment(x) \mtypeEnvironment(x)
\end{array} \end{array}
\end{displaymath} \end{displaymath}
\begin{displaymath} \begin{displaymath}
\begin{array}{@{}l@{}l} \begin{array}{@{}l@{}l}
\typeExpr{} &({\mtypeEnvironment} , \texttt{new}\ \type{C}(\overline{e}), \tv{a}) = \\ \typeExpr{} &({\mtypeEnvironment}, \Gamma , \texttt{new}\ \type{C}(\overline{e}), \tv{a}) = \\
& \begin{array}{ll} & \begin{array}{ll}
\textbf{let} \textbf{let}
& \ol{\ntv{r}}, \ol{\ntv{a}} \ \text{fresh} \\ & \ol{\ntv{r}}, \ol{\ntv{a}} \ \text{fresh} \\
& \overline{\consSet} = \typeExpr({\mtypeEnvironment}, \overline{e}, \ol{\ntv{r}}) \\ & \overline{\consSet} = \typeExpr({\mtypeEnvironment}, \Gamma, \overline{e}, \ol{\ntv{r}}) \\
& C = \set{\ol{\ntv{r}} \lessdot [\ol{\ntv{a}}/\ol{X}]\ol{T}, \ol{\ntv{a}} \lessdot \ol{N} \mid \texttt{class}\ \exptype{C}{\ol{X} \triangleleft \ol{N}} \set{ \ol{T\ f}; \ldots}} \\ & C = \set{\ol{\ntv{r}} \lessdot [\ol{\ntv{a}}/\ol{X}]\ol{T}, \ol{\ntv{a}} \lessdot \ol{N} \mid \texttt{class}\ \exptype{C}{\ol{X} \triangleleft \ol{N}} \set{ \ol{T\ f}; \ldots}} \\
{\mathbf{in}} & { {\mathbf{in}} & {
\overline{\consSet} \cup \overline{\consSet} \cup

34
helpers.tex Normal file
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@ -0,0 +1,34 @@
\section{Additional Functions}
\begin{description}
\item[$\tph{}$] returns all type placeholders inside a given type.
\textit{Example:} $\tph(\wctype{\wildcard{X}{\tv{a}}{\bot}}{Pair}{\wtv{b},\rwildcard{X}}) = \set{\tv{a}, \wtv{b}}$
%\textbf{Wildcard renaming}\\
\item[Wildcard renaming:]
The \rulename{Reduce} rule separates wildcards from their environment.
At this point each wildcard gets a new and unique name.
To only rename the respective wildcards the reduce rule renames wildcards up to alpha conversion:
($[\ol{C}/\ol{B}]$ in the \rulename{reduce} rule)
\begin{align*}
[\type{A}/\type{B}]\type{B} &= \type{A} \\
[\type{A}/\type{B}]\type{C} &= \type{C} & \text{if}\ \type{B} \neq \type{C}\\
[\type{A}/\type{B}]\wcNtype{\overline{\wildcard{X}{\type{U}}{\type{L}}}}{N} &= \wcNtype{\overline{\wildcard{X}{[\type{A}/\type{B}]\type{U}}{[\type{A}/\type{B}]\type{L}}}}{[\type{A}/\type{B}]N} & \text{if}\ \type{B} \notin \overline{\rwildcard{X}} \\
[\type{A}/\type{B}]\wcNtype{\overline{\wildcard{X}{\type{U}}{\type{L}}}}{N} &= \wcNtype{\overline{\wildcard{X}{\type{U}}{\type{L}}}}{N} & \text{if}\ \type{B} \in \overline{\rwildcard{X}} \\
\end{align*}
\item[Free Variables:]
The $\text{fv}$ function assumes every wildcard type variable to be a free variable aswell.
% TODO: describe a function which determines free variables? or do an example?
\begin{align*}
\text{fv}(\rwildcard{A}) &= \set{ \rwildcard{A} } \\
\text{fv}(\tv{a}) &= \emptyset \\
%\text{fv}(\wtv{a}) &= \set{\wtv{a}} \\
\text{fv}(\wctype{\Delta}{C}{\ol{T}}) &= \set{\text{fv}(\type{T}) \mid \type{T} \in \ol{T}} / \text{dom}(\Delta) \\
\end{align*}
\item[Fresh Wildcards:]
$\text{fresh}\ \overline{\wildcard{A}{\tv{u}}{\tv{l}}}$ generates fresh wildcards.
The new names $\ol{A}$ are fresh, aswell as the type variables $\ol{\tv{u}}$ and $\ol{\tv{l}}$,
which are used for the upper and lower bounds.
\end{description}

341
introduction.tex
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@ -383,344 +383,3 @@ let l1' : (*@$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$@*) =
% Polymorphic method calls need to be wrapped in a process which \textit{opens} existential types \cite{addingWildcardsToJava}. % Polymorphic method calls need to be wrapped in a process which \textit{opens} existential types \cite{addingWildcardsToJava}.
% In Java this is done implicitly in a process called capture conversion (as proposed in Wild FJ \cite{WildFJ}). % In Java this is done implicitly in a process called capture conversion (as proposed in Wild FJ \cite{WildFJ}).
\section{Global Type Inference Algorithm}
\label{sec:glob-type-infer}
This section gives an overview of the global type inference algorithm
with examples and identifies the challenges that we had to overcome in
the design of the algorithm.
\begin{figure}[h]
\begin{minipage}{0.49\textwidth}
\begin{lstlisting}[style=tfgj, caption=Valid Java program, label=lst:addExample]
<A> List<A> add(List<A> l, A v)
List<? super String> l = ...;
add(l, "String");
\end{lstlisting}
\end{minipage}\hfill
\begin{minipage}{0.49\textwidth}
\begin{lstlisting}[style=tamedfj, caption=\TamedFJ{} representation, label=lst:addExampleLet]
<A> List<A> add(List<A> l, A v)
List<? super String> l = ...;
let v:(*@$\tv{v}$@*) = l
in add(v, "String");
\end{lstlisting}
\end{minipage}\\
\begin{minipage}{0.49\textwidth}
\begin{lstlisting}[style=constraints, caption=Constraints, label=lst:addExampleCons]
(*@$\wctype{\wildcard{X}{\type{String}}{\bot}}{List}{\rwildcard{X}} \lessdot \tv{v}$@*)
(*@$\tv{v} \lessdotCC \exptype{List}{\wtv{a}}$@*)
(*@$\type{String} \lessdot \wtv{a}$@*)
\end{lstlisting}
\end{minipage}\hfill
\begin{minipage}{0.49\textwidth}
\begin{lstlisting}[style=letfj, caption=Type solution, label=lst:addExampleSolution]
<A> List<A> add(List<A> l, A v)
List<? super String> l = ...;
let l2:(*@$\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}}$@*) = l
in <X>add(l2, "String");
\end{lstlisting}
\end{minipage}
\end{figure}
% \begin{description}
% \item[input] \tifj{} program
% \item[output] type solution
% \item[postcondition] the type solution applied to the input must yield a valid \letfj{} program
% \end{description}
%Our algorithm is an extension of the \emph{Global Type Inference for Featherweight Generic Java}\cite{TIforFGJ} algorithm.
Listings~\ref{lst:addExample}, \ref{lst:addExampleLet}, \ref{lst:addExampleCons}, and
\ref{lst:addExampleSolution} showcase the global type inference algorithm step by step.
In the Java code in Listing~\ref{lst:addExample}, the type of variable
\texttt{l} is an existential type so that it has to undergo a capture conversion
before being passed to a method call.
To this end we convert the program to A-Normal form (Listing~\ref{lst:addExampleLet}),
which introduces a let statement defining a new variable \texttt{v}.
As the type of \texttt{v} is unknown, the algorithm puts a type
placeholder $\tv{v}$ for the type of \texttt{v} before generating constraints (see Listing~\ref{lst:addExampleCons}).
These constraints mirror the typing rules of the \TamedFJ{} calculus (section~\ref{sec:tifj}).
% During the constraint generation step the type of the variable \texttt{v} is unknown
% and given the type placeholder $\tv{v}$.
The call to \texttt{add} generates the \emph{capture constraint} $\tv{v} \lessdotCC \exptype{List}{\wtv{a}}$.
This constraint ($\lessdotCC$) is a kind of subtype constraint, which additionally
expresses that the left side of the constraint is subject to a capture conversion.
Next, the unification algorithm \unify{} (section~\ref{sec:unify}) solves the constraints.
Initially, no type is assigned to $\tv{v}$.
% During the course of \unify{} more and more types emerge.
As soon as a concrete type for $\tv{v}$ is appears during constraint
solving, \unify{} can conduct a capture conversion if needed.
%The constraints where this is possible are marked as capture constraints.
In this example, $\tv{v}$ will be set to
$\wctype{\wildcard{X}{\type{String}}{\bot}}{List}{\rwildcard{X}}$
leaving us with the following constraints:
\begin{displaymath}
\prftree[r]{Capture}{
\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}, \, \type{String} \lessdotCC \wtv{a}
}{
\wildcard{Y}{\type{Object}}{\type{String}} \wcSep \exptype{List}{\rwildcard{Y}} \lessdot \exptype{List}{\wtv{a}}, \, \type{String} \lessdot \wtv{a}
}
\end{displaymath}
%Capture Constraints $(\wctype{\rwildcard{X}}{C}{\rwildcard{X}} \lessdotCC \type{T})$ allow for a capture conversion,
%which converts a constraint of the form $(\wctype{\rwildcard{X}}{C}{\rwildcard{X}} \lessdotCC \type{T})$ to $(\exptype{C}{\rwildcard{X}} \lessdot \type{T})$
The constraint $\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}$
allows \unify{} to do a capture conversion to $\exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$.
The captured wildcard $\rwildcard{X}$ gets a fresh name \texttt{Y}, which is stored in the wildcard environment of the \unify{} algorithm.
Substituting \texttt{Y} for $\wtv{a}$ yields the constraints almost
solved: $\exptype{List}{\rwildcard{Y}} \lessdot
\exptype{List}{\rwildcard{Y}}$, $\type{String} \lessdot
\rwildcard{Y}$.
The first constraint is obviously satisfied and $\type{String} \lessdot \rwildcard{Y}$ is satisfied
because $\rwildcard{Y}$ has $\type{String}$ as lower bound.
A correct Featherweight Java program including all type annotations and an explicit capture conversion via let statement is shown in Listing \ref{lst:addExampleSolution}.
This program can be deduced from the solution of the \unify{} algorithm presented in Section~\ref{sec:unify}.
In the body of the let statement the type $\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}}$
becomes $\exptype{List}{\rwildcard{X}}$ and the capture converted wildcard $\wildcard{X}{\type{Object}}{\type{String}}$ can be used as
a type parameter of the call \texttt{<X>add(v, "String")}.
% The input to our type inference algorithm is a modified version of the calculus in \cite{WildcardsNeedWitnessProtection} (see chapter \ref{sec:tifj}).
% First \fjtype{} (see section \ref{chapter:constraintGeneration}) generates constraints
% and afterwards \unify{} (section \ref{sec:unify}) computes a solution for the given constraint set.
% Constraints consist out of subtype constraints $(\type{T} \lessdot \type{T})$ and capture constraints $(\type{T} \lessdotCC \type{T})$.
% \textit{Note:} a type $\type{T}$ can either be a named type, a type placeholder or a wildcard type placeholder.
% A subtype constraint is satisfied if the left side is a subtype of the right side according to the rules in figure \ref{fig:subtyping}.
% \textit{Example:} $\exptype{List}{\ntv{a}} \lessdot \exptype{List}{\type{String}}$ is fulfilled by replacing type placeholder $\ntv{a}$ with the type $\type{String}$.
% Subtype constraints and type placeholders act the same as the ones used in \emph{Type Inference for Featherweight Generic Java} \cite{TIforFGJ}.
% The novel capture constraints and wildcard placeholders are needed for method invocations involving wildcards.
% The central piece of this type inference algorithm, the \unify{} process, is described with implication rules (chapter \ref{sec:unify}).
% We try to keep the branching at a minimal amount to improve runtime behavior.
% Also the transformation steps of the \unify{} algorithm are directly related to the subtyping rules of our calculus.
% There are no informal parts in our \unify{} algorithm.
% It solely consist out of transformation rules which are bound to simple checks.
\subsection{Challenges}\label{challenges}
%TODO: Wildcard subtyping is infinite see \cite{TamingWildcards}
% Wildcards are not reflexive.
% ( on the equals property ), every wildcard has to be capture converted when leaving its scope
% do not substitute free type variables
Global type inference for Featherweight Generic Java without wildcards
has been considered elsewher \cite{TIforFGJ}.
Adding wildcards to the calculus created new problems, which have not
been recognized by existing work on type unification for Java with wildcards~\cite{plue09_1}.
% what is the problem?
% Java does invisible capture conversion steps
Java's wildcard types are represented as existential types that have to be opened before they can be used.
Opening can either be done implicitly
(\cite{aModelForJavaWithWildcards}, \cite{javaTIisBroken}) or
explicitly via a let statement (\cite{WildcardsNeedWitnessProtection}).
For all variations it is vital to know the argument types with which a method is called.
In the absence of type annotations,
we do not know where an existential type will emerge and where a capture conversion is necessary.
Rather, the type inference algorithm has to figure out the placement
of existentials.
We identified three main challenges related to Java wildcards and global type inference.
\begin{enumerate}
\item \label{challenge:1}
One challenge is to design the algorithm in a way that it finds a
correct solution for programs like the one shown in Listing~\ref{lst:addExample}
and rejects programs like the one in Listing~\ref{lst:concatError}.
The first one is a valid Java program,
because the type \texttt{List<? super String>} is \textit{capture converted} to a fresh type variable $\rwildcard{X}$
which is used as the generic method parameter for the call to \texttt{add} as shown in Listing~\ref{lst:addExampleLet}.
Because we know that the type \texttt{String} is a subtype of the free variable $\rwildcard{X}$,
it is safe to pass \texttt{"String"} for the first parameter of the function.
The second program shown in Listing~\ref{lst:concatError} is incorrect.
The method call to \texttt{concat} with two wildcard lists is unsound.
The element types of the lists may be unrelated, therefore the call to \texttt{concat} cannot succeed.
The problem gets apparent if we try to write the \texttt{concat}
method call in the \TamedFJ{} calculus
(Listing~\ref{lst:concatTamedFJ}):
\texttt{l1'} and \texttt{l2'} are two different lists inside the body of the let statements, namely
$\exptype{List}{\rwildcard{X}}$ and $\exptype{List}{\rwildcard{Y}}$.
For the method call \texttt{concat(x1, x2)} no replacement for the generic \texttt{A}
exists to satisfy
$\exptype{List}{\type{A}} <: \exptype{List}{\type{X}},
\exptype{List}{\type{A}} <: \exptype{List}{\type{Y}}$.
% \textbf{Solution:}
% Capture Conversion during Unify.
\item
\unify{} transforms a constraint set into a correct type solution by
gradually assigning types to type placeholders during that process.
Solved constraints are removed and never considered again.
In the following example, \unify{} solves the constraint generated by the expression
\texttt{l.add(l.head())} first, which results in $\ntv{l} \lessdot \exptype{List}{\wtv{a}}$.
\begin{verbatim}
anyList() = new List<String>() :? new List<Integer>()
add(anyList(), anyList().head());
\end{verbatim}
The type for \texttt{l} can be an arbitrary list, but it has to be a invariant one.
Assigning a \texttt{List<?>} for \texttt{l} is unsound, because the type list hiding behind
\texttt{List<?>} could be a different one for the \texttt{add} call than the \texttt{head} method call.
An additional constraint $\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$
is solved by removing the wildcard $\rwildcard{X}$ if possible.
This problem is solved by ANF transformation
\item
% This problem is solved by assuming everything is a wildcard and lateron erasing excessive wildcards
% this is solved by having wildcards bound to a type. But this makes it necessary to remove wildcards lateron otherwise Unify would have to backtrack
The program in Listing~\ref{shuffleExample} exhibits a challenge involving wildcards and subtyping.
The method call \texttt{shuffle(l)} is incorrect, because \texttt{l} has type
$\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$ representing a list of unknown lists.
However, $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$ is a subtype of
$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$ which represents a list of lists, all of the same type $\rwildcard{X}$,
and can be passed safely to \texttt{shuffle}.
This behavior can also be explained by looking at the types and their capture converted versions:
\begin{center}
\begin{tabular}{l | l | l}
Java type & \TamedFJ{} representation & Capture Conversion \\
\hline
$\exptype{List}{\exptype{List}{\texttt{?}}}$ & $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$ & $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$ \\
$\exptype{List2D}{\texttt{?}}$ & $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$ & $\exptype{List2D}{\rwildcard{X}}$ \\
%Supertype of $\exptype{List2D}{\texttt{?}}$ & $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$ & $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$ \\
\end{tabular}
\end{center}
%The direct supertype of $\exptype{List2D}{\rwildcard{X}}$ is $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$.
%One problem is the divergence between denotable and expressable types in Java \cite{semanticWildcardModel}.
%A wildcard in the Java syntax has no name and is bound to its enclosing type:
%$\exptype{List}{\exptype{List}{\type{?}}}$ equates to $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$.
%During type checking \emph{intermediate types}
%can emerge, which have no equivalent in the Java syntax.
\begin{lstlisting}[style=java,label=shuffleExample,caption=Intermediate Types Example]
class List<X> extends Object {...}
class List2D<X> extends List<List<X>> {...}
<X> void shuffle(List<List<X>> list) {...}
List<List<?>> l = ...;
List2D<?> l2d = ...;
shuffle(l); // Error
shuffle(l2d); // Valid
\end{lstlisting}
% Java is using local type inference to allow method invocations which are not describable with regular Java types.
% The \texttt{shuffle} method in this case is invoked with the type $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$
% which is a subtype of $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$.
% After capture conversion \texttt{l2d'} has the type $\exptype{List}{\exptype{List}{\rwildcard{X}}}$
% and \texttt{shuffle} can be invoked with the type parameter $\rwildcard{X}$:
% \begin{lstlisting}[style=TamedFJ]
% let l2d' : (*@$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$@*) = l2d in <X>shuffle(l2d')
% \end{lstlisting}
% For the example shown in listing \ref{shuffleExample} the method call \texttt{shuffle(l2d)} creates the constraints:
% \begin{constraintset}
% \begin{center}
% $
% \begin{array}{l}
% \wctype{\rwildcard{X}}{List2D}{\rwildcard{X}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}
% \\
% \hline
% \wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}
% \\
% \hline
% \textit{Capture Conversion:}\
% \exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\wtv{x}}}
% \\
% \hline
% \textit{Solution:} \wtv{x} \doteq \rwildcard{X} \implies \exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\rwildcard{X}}}
% \end{array}
% $
% \end{center}
% \end{constraintset}
Given such a program the type inference algorithm has to allow the call \texttt{shuffle(l2d)} and
decline the call to \texttt{shuffle(l)}.
% The method call \texttt{shuffle(l)} is invalid however,
% because \texttt{l} has the type
% $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$.
% There is no solution for the subtype constraint:
% $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}$
\item \label{challenge3} \textbf{Free variables cannot leave their scope}:
Let's assume we have a variable \texttt{ls} with type $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$
%When calling the \texttt{id} function with an element of this list we have to apply capture conversion.
and the following program:
\noindent
\begin{minipage}{0.62\textwidth}
\begin{lstlisting}
let x : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) = ls.get(0) in id(x) : (*@$\ntv{z}$@*)
\end{lstlisting}\end{minipage}
\hfill
\begin{minipage}{0.36\textwidth}
\begin{lstlisting}[style=constraints]
(*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{x}}$@*),
(*@$\exptype{List}{\wtv{x}} \lessdot \ntv{z},$@*)
\end{lstlisting}
\end{minipage}
% the variable z has to
Take the Java program in listing \ref{lst:mapExample} for example.
It uses map to apply a polymorphic method \texttt{id} to every element of a list
$\expr{l} :
\exptype{List}{\wctype{\rwildcard{A}}{List}{\rwildcard{A}}}$.
How do we get the \unify{} algorithm to determine the correct type for the
variable \expr{l2}?
Although we do not specify constraint generation for language constructs like
lambda expressions used in this example,
we can imagine that the constraints have to look like in Listing~\ref{lst:mapExampleCons}.
\begin{minipage}{0.45\textwidth}
\begin{lstlisting}[caption=List Map Example,label=lst:mapExample]
<X> List<X> id(List<X> l){ ... }
List<List<?>> ls;
l2 = l.map(x -> id(x));
\end{lstlisting}\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
\begin{lstlisting}[style=constraints, caption=Constraints, label=lst:mapExampleCons]
(*@$\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\wtv{x}}$@*)
(*@$\exptype{List}{\wtv{x}} \lessdot \tv{z},$@*)
(*@$\exptype{List}{\tv{z}} \lessdot \tv{l2}$@*)
\end{lstlisting}
\end{minipage}
The constraints
$\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\wtv{x}},
\exptype{List}{\wtv{x}} \lessdot \tv{z}$
stem from the body of the lambda expression
\texttt{id(x)}.
\textit{For clarification:} This method call would be represented as the following expression in \letfj{}:
\texttt{let x1 :$\wctype{\rwildcard{A}}{List}{\rwildcard{A}}$ = x in id(x) :$\tv{z}$}
The T-Let rule prevents us from using free variables created by the method call to \expr{id}
to be used in the return type $\tv{z}$.
But this restriction has to be communicated to the \unify{} algorithm,
which does not know about the origin and context of
the constraints.
If we naively substitute $\sigma(\tv{z}) = \exptype{List}{\rwildcard{A}}$
the return type of the \texttt{map} function would be the type
$\exptype{List}{\exptype{List}{\rwildcard{A}}}$, which would be unsound.
The type of \expr{l2} is the same as the one of \expr{l}:
$\exptype{List}{\wctype{\rwildcard{A}}{List}{\rwildcard{A}}}$
\textbf{Solution:}
We solve this issue by distinguishing between wildcard placeholders
and normal placeholders and introducing them as needed.
$\ntv{z}$ is a normal placeholder and is not allowed to contain free variables.
\end{enumerate}
%TODO: Move this part. or move the third challenge some underneath.
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "TIforWildFJ"
%%% End:

578
soundness.tex
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@ -1,118 +1,109 @@
\section{Soundness} \section{Soundness}
\begin{lemma} % \begin{lemma}
A sound TypelessFJ program is also sound under LetFJ type rules. % A sound TypelessFJ program is also sound under LetFJ type rules.
\begin{description} % \begin{description}
\item[if:] % \item[if:]
$\Gamma | \Delta \vdash \texttt{m}(\ol{x}) = \texttt{e} \ \ok \ \text{in}\ C \text{with} \ \generics{\ol{Y \triangleleft P}}$ % $\Gamma | \Delta \vdash \texttt{m}(\ol{x}) = \texttt{e} \ \ok \ \text{in}\ C \text{with} \ \generics{\ol{Y \triangleleft P}}$
\end{description} % \end{description}
\end{lemma} % \end{lemma}
TODO: Beforehand we have to show that $\Delta \cup \overline{\Delta} | \Theta \vdash \texttt{e} : \type{T} \mid \overline{\Delta}$ % TODO: Beforehand we have to show that $\Delta \cup \overline{\Delta} | \Theta \vdash \texttt{e} : \type{T} \mid \overline{\Delta}$
Here $\Delta$ does not contain every $\overline{\Delta}$ ever created. % Here $\Delta$ does not contain every $\overline{\Delta}$ ever created.
%what prevents a free variable to emerge in \Delta.N example Y^Object |- C<String> <: X^Y.C<X> % %what prevents a free variable to emerge in \Delta.N example Y^Object |- C<String> <: X^Y.C<X>
% if the Y is later needed for an equals: same(id(x), x2) % % if the Y is later needed for an equals: same(id(x), x2)
Free wildcards do not move inwards. We can show that every new type is either well-formed and therefore does not contain any free variables. % Free wildcards do not move inwards. We can show that every new type is either well-formed and therefore does not contain any free variables.
Or it is a generic method call: is it possible to use any free wildcards here? % Or it is a generic method call: is it possible to use any free wildcards here?
let empty % let empty
<X> Box<X> empty() % <X> Box<X> empty()
same(Box<?>, empty()) % same(Box<?>, empty())
let p1 : X.Box<X> = Box<?> in let % let p1 : X.Box<X> = Box<?> in let
X.Box<X> <. Box<x> % X.Box<X> <. Box<x>
Box<e> <. Box<x> % Box<e> <. Box<x>
boxin(empty()), Box2<?> % boxin(empty()), Box2<?>
Where can a problem arise? When we use free wildcards before they are freed. % Where can a problem arise? When we use free wildcards before they are freed.
But we can always CC them first. Exception two types: X.Pair<X, y> and Y.Pair<x, Y> % But we can always CC them first. Exception two types: X.Pair<X, y> and Y.Pair<x, Y>
Here y = Y and x = X but % Here y = Y and x = X but
<X,Y> void same(Pair<X,Y> a, Pair<X,Y> b){} % <X,Y> void same(Pair<X,Y> a, Pair<X,Y> b){}
<X> Pair<?, X> left() { return null; } % <X> Pair<?, X> left() { return null; }
<X> Pair<X, ?> right() { return null; } % <X> Pair<X, ?> right() { return null; }
<X> Box<X> id(Box<? extends Box<X>> x) % <X> Box<X> id(Box<? extends Box<X>> x)
here it could be beneficial to use a free wildcard as the parameter X to have it later % here it could be beneficial to use a free wildcard as the parameter X to have it later
Box<?> x = ... % Box<?> x = ...
same(id(x), id(x)) <- this will be accepted by TI % same(id(x), id(x)) <- this will be accepted by TI
let left : X,Y.Pair<X,Y> = left() in % let left : X,Y.Pair<X,Y> = left() in
let right : Pair<X,Y> = right() in % let right : Pair<X,Y> = right() in
Compromise: % Compromise:
- Generate constraints so that they comply with LetFJ % - Generate constraints so that they comply with LetFJ
- Propose a version which is close to Java % - Propose a version which is close to Java
Version for LetFJ: % Version for LetFJ:
Is it still possible to do the capture conversion in form of constraints? % Is it still possible to do the capture conversion in form of constraints?
X.C<X> <. C<x> % X.C<X> <. C<x>
T <. X.C<X> % T <. X.C<X>
how to proof: X.C<X> ok % how to proof: X.C<X> ok
If $\Delta \cup \overline{\Delta} | \Theta \vdash \texttt{e} : \type{T} \mid \overline{\Delta}$ % If $\Delta \cup \overline{\Delta} | \Theta \vdash \texttt{e} : \type{T} \mid \overline{\Delta}$
then there exists a $|\texttt{e}|$ with $\Delta | \Theta \vdash |\texttt{e}| : \wcNtype{\Delta'}{N}$ in LetFJ. % then there exists a $|\texttt{e}|$ with $\Delta | \Theta \vdash |\texttt{e}| : \wcNtype{\Delta'}{N}$ in LetFJ.
This is possible by starting with the parameter types as the base case $\overline{\Delta} = \emptyset$. % This is possible by starting with the parameter types as the base case $\overline{\Delta} = \emptyset$.
Each type $\wcNtype{\Delta'}{N}$ can only use wildcards already freed. % Each type $\wcNtype{\Delta'}{N}$ can only use wildcards already freed.
\textit{Proof} by structural induction. % \textit{Proof} by structural induction.
\begin{description} % \begin{description}
\item[$\texttt{e} = \texttt{x}$] $\Delta | \Theta \vdash \texttt{e} : \type{T} \mid \emptyset$ % \item[$\texttt{e} = \texttt{x}$] $\Delta | \Theta \vdash \texttt{e} : \type{T} \mid \emptyset$
$\Delta \vdash \type{T} \ \ok$ by \rulename{T-Method} % $\Delta \vdash \type{T} \ \ok$ by \rulename{T-Method}
and therefore $\Delta | \Theta \vdash \texttt{let}\ \texttt{e} : \type{T} = \texttt{x in } \texttt{e}$. % and therefore $\Delta | \Theta \vdash \texttt{let}\ \texttt{e} : \type{T} = \texttt{x in } \texttt{e}$.
$|\texttt{x}, \texttt{e}| = \texttt{let}\ \texttt{e} : \type{T} = \texttt{x in } \texttt{e}$ % $|\texttt{x}, \texttt{e}| = \texttt{let}\ \texttt{e} : \type{T} = \texttt{x in } \texttt{e}$
\item[$\texttt{e} = \texttt{e}.\texttt{m}(\ol{e})$] there must be atleast one value in $\texttt{e}$ or $\ol{e}$ % \item[$\texttt{e} = \texttt{e}.\texttt{m}(\ol{e})$] there must be atleast one value in $\texttt{e}$ or $\ol{e}$
\item[$\texttt{e}.f$] given let x : T = e' in x % \item[$\texttt{e}.f$] given let x : T = e' in x
let x : T = e' in let xf = x.f in xf % let x : T = e' in let xf = x.f in xf
Required: % Required:
$ \Delta | \Theta \vdash e' : \type{T}_1$ % $ \Delta | \Theta \vdash e' : \type{T}_1$
$\Delta \vdash \type{T}_1 <: \wcNtype{\Delta'}{N}$ % $\Delta \vdash \type{T}_1 <: \wcNtype{\Delta'}{N}$
$\Delta, \Delta' | \Theta, x : \type{N} \vdash let xf = x.f in xf : \type{T}_2$ % $\Delta, \Delta' | \Theta, x : \type{N} \vdash let xf = x.f in xf : \type{T}_2$
\end{description} % \end{description}
\textbf{Proof:} Every program complying with our type rules can be converted to a correct LetFJ program. % \textbf{Proof:} Every program complying with our type rules can be converted to a correct LetFJ program.
First we convert the program so that every wildcards used in an expression are in the $\Delta$ environment: % First we convert the program so that every wildcards used in an expression are in the $\Delta$ environment:
m(p) = e => let xp = p in [xp/p]e % m(p) = e => let xp = p in [xp/p]e
x1.m(x2) => let xm = x1.m(x2=) in xm % x1.m(x2) => let xm = x1.m(x2=) in xm
x.f => let xf = x.f in xf % x.f => let xf = x.f in xf
Then we have to proof there is never a wildcard used before it is declared. % Then we have to proof there is never a wildcard used before it is declared.
Wildcards are introduced by the capture conversions and nothing else. % Wildcards are introduced by the capture conversions and nothing else.
\begin{theorem}\label{testenv-theorem} % \begin{lemma}{Well-formedness:}
Type inference produces a correctly typed program. % TODO:
\begin{description} % \end{lemma}
\item[If] $\fjtypeinference(\mv{\Pi}, \texttt{class}\ \exptype{C}{\ol{X}
\triangleleft \ol{N}} \triangleleft \type{N}\ \{ \overline{\type{T} \ f};\ \ol{M} \}) = \mtypeEnvironment{}'$
\item[Then] $\texttt{class}\ \exptype{C}{\ol{X}
\triangleleft \ol{N}} \triangleleft \type{N}\ \{ \overline{\type{T} \ f};\ \ol{M} \} \text{ok}$,
with $\ol{M} = $
\end{description}
\end{theorem}
\begin{lemma}{Well-formedness:}
TODO:
\end{lemma}
\begin{lemma}{Soundness:} \begin{lemma}{Soundness:}
\unify{}'s type solutions for a constraint set generated by $\typeExpr{}$ are correct. \unify{}'s type solutions for a constraint set generated by $\typeExpr{}$ are correct.
\begin{description} \begin{description}
\item[if] $\typeExpr{}(\mtypeEnvironment{}, \texttt{e}, \tv{a}) = C$ \item[if] $\typeExpr{}(\mtypeEnvironment{}, \texttt{e}, \tv{a}) = (\Delta', C)$
and $(\Delta_u, \sigma) = \unify{}(\Delta', C)$ and let $\Delta = \Delta_u \cup \Delta'$ and $(\Delta_u, \sigma) = \unify{}(\Delta', C)$ and let $\Delta= \Delta_u \cup \Delta'$
$\Delta, \Delta' \vdash $
% , with $C = \set{ \overline{ \type{S} \lessdot \type{T} } \cup \overline{ \type{S'} \lessdotCC \type{T'} } }$ % , with $C = \set{ \overline{ \type{S} \lessdot \type{T} } \cup \overline{ \type{S'} \lessdotCC \type{T'} } }$
% and $\vdash \ol{L} : \mtypeEnvironment{}$ % and $\vdash \ol{L} : \mtypeEnvironment{}$
% and $\Gamma \subseteq \mtypeEnvironment{}$ % and $\Gamma \subseteq \mtypeEnvironment{}$
% \item[given] a $(\Delta, \sigma)$ with $\Delta \vdash \overline{\sigma(\type{S}) <: \sigma(\type{T})}$ % \item[given] a $(\Delta, \sigma)$ with $\Delta \vdash \overline{\sigma(\type{S}) <: \sigma(\type{T})}$
% and there exists a $\Delta'$ with $\Delta, \Delta' \vdash \overline{\CC{}(\sigma(\type{S'})) <: \sigma(\type{T'})}$ % and there exists a $\Delta'$ with $\Delta, \Delta' \vdash \overline{\CC{}(\sigma(\type{S'})) <: \sigma(\type{T'})}$
\item[then] there is a completion $|\texttt{e}|$ with $\Delta|\Gamma \vdash |\texttt{e}| : \sigma(\tv{a})$ %\item[then] there is a completion $|\texttt{e}|$ with $\Delta|\Gamma \vdash |\texttt{e}| : \sigma(\tv{a})$
\item[then] $\Delta|\Gamma \vdash \texttt{e} : \sigma(\tv{a})$
\end{description} \end{description}
\end{lemma} \end{lemma}
Regular type placeholders represent type annotations. Regular type placeholders represent type annotations.
@ -129,6 +120,16 @@ used in let statements and as type parameters for generic method calls.
\textit{Proof:} \textit{Proof:}
By structural induction over the expression $\texttt{e}$. By structural induction over the expression $\texttt{e}$.
\begin{description} \begin{description}
\item[$\texttt{let}\ \texttt{x} = \texttt{t}_1 \ \texttt{in}\ \expr{t}_2$]
$\Delta|\Gamma \vdash \expr{t}_1: \type{T}_1$ by assumption.
The body of the let statement will only use the free variables provided by $\Delta$ and $\Delta'$ (see lemma \ref{lemma:wildcardsStayInScope}).
\item[$\expr{v}.\texttt{f}$]
We are alowwed to use capture conversion for $\expr{v}$ here.
\item[$\texttt{let}\ \texttt{x} = \texttt{e} \ \texttt{in} \ \texttt{x}.\texttt{f}$]
$\Delta|\Gamma \vdash \expr{e}: \type{T}$ by assumption.
$\text{dom}(\Delta') \subseteq \text{fv}(\type{N})$ by lemma \ref{lemma:wildcardWellFormedness}.
$\Delta, \Delta' | \Gamma, \expr{x} : \type{T} \vdash \texttt{x}.\texttt{f}$
\item[$\texttt{e}.\texttt{f}$] Let $\sigma(\tv{r}) = \wcNtype{\Delta_c}{N}$, \item[$\texttt{e}.\texttt{f}$] Let $\sigma(\tv{r}) = \wcNtype{\Delta_c}{N}$,
then $\Delta|\Gamma \vdash \texttt{e} : \wcNtype{\Delta_c}{N}$ by assumption. then $\Delta|\Gamma \vdash \texttt{e} : \wcNtype{\Delta_c}{N}$ by assumption.
$\Delta', \Delta, \Delta_c \vdash \type{N} <: \sigma(\exptype{C}{\overline{\wtv{a}}})$ by premise. $\Delta', \Delta, \Delta_c \vdash \type{N} <: \sigma(\exptype{C}{\overline{\wtv{a}}})$ by premise.
@ -157,28 +158,29 @@ By structural induction over the expression $\texttt{e}$.
% This can be given by the constraints generated. We can proof if T ok and S <: T and T <: S' then S ok and S' ok % This can be given by the constraints generated. We can proof if T ok and S <: T and T <: S' then S ok and S' ok
% If S ok and T <. S , then Unify generates a T ok % If S ok and T <. S , then Unify generates a T ok
S typeinference:
T <: [S/Y]U
We apply the following lemma
Lemma
if T ok and T <: S then S ok
until % S typeinference:
T = [S/Y]U % T <: [S/Y]U
% We apply the following lemma
% Lemma
% if T ok and T <: S then S ok
and then we can say by % until
Lemma: % T = [S/Y]U
If [S/Y]U ok then S ok (TODO: proof!)
So we do not have to proof S ok (but T) % and then we can say by
% Lemma:
% If [S/Y]U ok then S ok (TODO: proof!)
% T_r <: C<T> (S is in T) % So we do not have to proof S ok (but T)
% Is C<T> ok?
% if every type environment \Delta supplied to Unify is ok (L <: U), then \sigma(a) = \Delta'.N implies \Delta' conforms to (L <: U)
% this together with the X <. N constraints proofs T_r ok
$\Delta \vdash \sigma(\tv{a}), \wcNtype{\Delta_c}{N} \ \ok$ %TODO % % T_r <: C<T> (S is in T)
%Easy, because unify only generates substitutions for normal type placeholders which are OK % % Is C<T> ok?
% % if every type environment \Delta supplied to Unify is ok (L <: U), then \sigma(a) = \Delta'.N implies \Delta' conforms to (L <: U)
% % this together with the X <. N constraints proofs T_r ok
% $\Delta \vdash \sigma(\tv{a}), \wcNtype{\Delta_c}{N} \ \ok$ %TODO
% %Easy, because unify only generates substitutions for normal type placeholders which are OK
\item[$\texttt{e}.\texttt{m}(\ol{e})$] \item[$\texttt{e}.\texttt{m}(\ol{e})$]
Lets have a look at the case where the receiver and parameter types are all named types. Lets have a look at the case where the receiver and parameter types are all named types.
@ -194,7 +196,8 @@ $\texttt{let}\ \texttt{x} : \wcNtype{\Delta'}{N} = \texttt{e} \ \texttt{in}\
% Solution: Make this a lemma and guarantee that Unify does only create types Delta.T with \Delta subset fv(T) % Solution: Make this a lemma and guarantee that Unify does only create types Delta.T with \Delta subset fv(T)
% This is possible because free variables (except those in \Delta_in) are never used in wildcard environments % This is possible because free variables (except those in \Delta_in) are never used in wildcard environments
By lemma \ref{lemma:unifyWeakening} we know that %By lemma \ref{lemma:unifyWeakening}
We know that
$\unify{}(\Delta, [\ol{\tv{a}}/\ol{X}][\ol{\tv{b}}/\ol{Y}]\set{ \overline{\wcNtype{\Delta}{N}} \lessdotCC \ol{T}, \wcNtype{\Delta'}{N} \lessdotCC \exptype{C}{\ol{X}}})$ $\unify{}(\Delta, [\ol{\tv{a}}/\ol{X}][\ol{\tv{b}}/\ol{Y}]\set{ \overline{\wcNtype{\Delta}{N}} \lessdotCC \ol{T}, \wcNtype{\Delta'}{N} \lessdotCC \exptype{C}{\ol{X}}})$
has a solution. has a solution.
Also $\overline{\wcNtype{\Delta}{N}} \lessdot \ol{T}$ and $\wcNtype{\Delta'}{N} \lessdotCC \exptype{C}{\ol{X}}$ will be converted to Also $\overline{\wcNtype{\Delta}{N}} \lessdot \ol{T}$ and $\wcNtype{\Delta'}{N} \lessdotCC \exptype{C}{\ol{X}}$ will be converted to
@ -206,12 +209,13 @@ which shows $\Delta, \Delta', \overline{\Delta} \vdash \overline{N} <: [\ol{S}/\
$\Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} \type{T}$, $\Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} \type{T}$,
$\Delta, \Delta', \overline{\Delta} \vdash \ol{N} <: [\ol{S}/\ol{X}]\ol{U}$, $\Delta, \Delta', \overline{\Delta} \vdash \ol{N} <: [\ol{S}/\ol{X}]\ol{U}$,
and $\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}$ and $\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}$
are guaranteed by the generated constraints and lemma \ref{lemma:unifySoundness} and \ref{lemma:unifyCC}. are guaranteed by the generated constraints and lemma \ref{lemma:unifySoundness}.% and \ref{lemma:unifyCC}.
$\Delta, \Delta' \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}$ due to $\ol{T} = \ol{\wcNtype{\Delta}{N}}$ and S-Refl. $\Delta, \Delta' \vdash \type{T}_r <: \wcNtype{\Delta'}{N}$ accordingly. $\Delta, \Delta' \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}$ due to $\ol{T} = \ol{\wcNtype{\Delta}{N}}$ and S-Refl. $\Delta, \Delta' \vdash \type{T}_r <: \wcNtype{\Delta'}{N}$ accordingly.
$\Delta|\Gamma \vdash \texttt{t}_r : \type{T}_r$ and $\Delta|\Gamma \vdash \ol{t} : \ol{T}_r$ by assumption. $\Delta|\Gamma \vdash \texttt{t}_r : \type{T}_r$ and $\Delta|\Gamma \vdash \ol{t} : \ol{T}_r$ by assumption.
$\Delta, \Delta' \vdash \ol{\wcNtype{\Delta}{N}} \ \ok$ by lemma \ref{lemma:unifyWellFormedness}, $\Delta, \Delta' \vdash \ol{\wcNtype{\Delta}{N}} \ \ok$ by lemma \ref{lemma:unifyWellFormedness},
therefore $\Delta, \Delta', \overline{\Delta} \vdash \ol{N} \ \ok$, which implies $\Delta, \Delta', \overline{\Delta} \vdash \ol{U} \ \ok$ therefore $\Delta, \Delta', \overline{\Delta} \vdash \ol{N} \ \ok$, which implies $\Delta, \Delta', \overline{\Delta} \vdash \ol{U} \ \ok$
by lemma \ref{lemma:wfHereditary} and $\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok $ by premise of WF-Class. %by lemma \ref{lemma:wfHereditary}
and $\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok $ by premise of WF-Class.
The same goes for $\wcNtype{\Delta'}{N}$. The same goes for $\wcNtype{\Delta'}{N}$.
% \begin{gather} % \begin{gather}
@ -270,29 +274,45 @@ by induction over every step of the \unify{} algorithm.
Hint: a type placeholder $\tv{a}$ will never be replaced by a free variable or a type containing free variables. Hint: a type placeholder $\tv{a}$ will never be replaced by a free variable or a type containing free variables.
This fact together with the presumption that every supplied type is well-formed we can easily show that this lemma is true. This fact together with the presumption that every supplied type is well-formed we can easily show that this lemma is true.
\textit{Proof: (Variant 2)} % \textit{Proof: (Variant 2)}
The GenSigma and GenDelta rules check for well-formedness. % The GenSigma and GenDelta rules check for well-formedness.
\begin{lemma}\label{lemma:wildcardWellFormedness}
\unify{} generates well-formed existential types
\begin{description}
\item[If] $(\Delta, \sigma) = \unify{}(\Delta', C)$
\item[and] $\forall \wcNtype{\Delta''}{N} \in C: \text{dom}(\Delta') \subseteq \text{fv}(\type{N})$
\item[and] $\sigma(\tv{a}) = \wcNtype{\Delta'}{N}$
%\item[then] $\Delta \vdash \ol{L <: U}$ and $\Delta \vdash \ol{U <: U'}$ % (with U' being upper limit by class definition)
\item[then] $\text{dom}(\Delta') \subseteq \text{fv}(\type{N})$
\end{description}
\end{lemma}
\textit{Proof} is straightforward induction over the transformation rules of \unify{}
under the assumption that all supplied existential types are satisfying the premise.
All parts of \unify{} that generate wildcard types always spawn types $\wcNtype{\Delta'}{N}$
with $\text{dom}(\Delta') \subseteq \text{fv}(\type{N})$.
% All types supplied to the Unify algorithm by TYPEs only contain ? extends or ? super wildcards. (X : [bot..T] or X : [T .. Object]). % All types supplied to the Unify algorithm by TYPEs only contain ? extends or ? super wildcards. (X : [bot..T] or X : [T .. Object]).
% Both are well formed! % Both are well formed!
\begin{lemma}\label{lemma:wfHereditary} % \begin{lemma}\label{lemma:wfHereditary}
Well-formedness is hereditary % Well-formedness is hereditary
\begin{description} % \begin{description}
\item[If] $\triangle \vdash \exptype{C}{\ol{X}} \ \ok$ and $\triangle \vdash \exptype{C}{\ol{X}} <: \type{S}$ % \item[If] $\triangle \vdash \exptype{C}{\ol{X}} \ \ok$ and $\triangle \vdash \exptype{C}{\ol{X}} <: \type{S}$
\item[Then] $\triangle \vdash \type{S} \ \ok$ % \item[Then] $\triangle \vdash \type{S} \ \ok$
\end{description} % \end{description}
\end{lemma} % \end{lemma}
\textit{Proof:} % \textit{Proof} by induction on subtyping rules in figure \ref{fig:subtyping}
\begin{lemma} % \begin{lemma}
\begin{description} % \begin{description}
\item[If] $\Delta \vdash \wctype{\Delta'}{C}{\ol{T}} \ \ok$ % \item[If] $\Delta \vdash \wctype{\Delta'}{C}{\ol{T}} \ \ok$
\item[Then] $\Delta, \Delta' \vdash \ok{T} \ \ok$ % \item[Then] $\Delta, \Delta' \vdash \ok{T} \ \ok$
\end{description} % \end{description}
\end{lemma} % \end{lemma}
\textit{Proof:} by definition of WF-Class % \textit{Proof:} by definition of WF-Class
\begin{lemma} \label{lemma:tvsNoFV} \begin{lemma} \label{lemma:tvsNoFV}
@ -313,16 +333,30 @@ Trivial. \unify{} fails when a constraint $\tv{a} \doteq \rwildcard{X}$ arises.
% Delta |- N <. T % Delta |- N <. T
% \end{lemma} % \end{lemma}
\begin{lemma} \label{lemma:unifyWeakening} \begin{lemma}\label{lemma:wildcardsStayInScope}
Removing constraints does not render \unify{} impossible as long as the removed constraints do not share wildcard placeholders. Free variables can only hop to another constraint if they share the same wildcard placeholder.
If there is a solution for a constraint set $C$, then there is also a solution for a subset of that constraint set.
\begin{description} \begin{description}
\item[If] $(\sigma, \Delta) = \unify{}( C \cup C')$ and $\text{wtv}(C) \cap \text{wtv}(C') = \emptyset$ \item[If] $(\sigma, \Delta) = \unify{} (\Delta', C \cup \set{\wcNtype{\Delta'}{N} \lessdot \tv{a}})$
\item[then] $ (\sigma', \Delta') = \unify{}( C')$ \item[and] $\tv{a}$ being a type placeholders used in $C$
\item[then] $\text{fv}(\sigma(\tv{a})) \subseteq \Delta, \Delta'$
\end{description} \end{description}
If $\wcNtype{\Delta'}{N} \lessdot \tv{a}$
\end{lemma} \end{lemma}
\textit{Proof:}
%TODO: use this to proof soundness. Wildcard placeholders are not shared with other constraints and therefore only the wildcards generated by capture conversion are used in a let statement.
% \begin{lemma} \label{lemma:unifyWeakening}
% Removing constraints does not render \unify{} impossible as long as the removed constraints do not share wildcard placeholders.
% If there is a solution for a constraint set $C$, then there is also a solution for a subset of that constraint set.
% \begin{description}
% \item[If] $(\sigma, \Delta) = \unify{}( C \cup C')$% and $\text{wtv}(C) \cap \text{wtv}(C') = \emptyset$
% \item[then] $ (\sigma', \Delta') = \unify{}( C')$
% \end{description}
% \end{lemma}
%\textit{Proof:}
%TODO %TODO
% does this really work. We have to show that the algorithm never gets stuck as long as there is a solution % does this really work. We have to show that the algorithm never gets stuck as long as there is a solution
% maybe there are substitutions with types containing free variables that make additional solutions possible. Check! % maybe there are substitutions with types containing free variables that make additional solutions possible. Check!
@ -507,158 +541,158 @@ Same as Subst
\end{description} \end{description}
\subsection{Converting to Wild FJ} % \subsection{Converting to Wild FJ}
Wildcards are existential types which have to be \textit{unpacked} before they can be used. % Wildcards are existential types which have to be \textit{unpacked} before they can be used.
In Java this is done implicitly by a process called capture conversion \cite{JavaLanguageSpecification}. % In Java this is done implicitly by a process called capture conversion \cite{JavaLanguageSpecification}.
The type system in \cite{WildcardsNeedWitnessProtection} makes this process explicit by using \texttt{let} statements. % The type system in \cite{WildcardsNeedWitnessProtection} makes this process explicit by using \texttt{let} statements.
Our type inference algorithm will accept an input program without let statements and add them where necessary. % Our type inference algorithm will accept an input program without let statements and add them where necessary.
%Type inference adds \texttt{let} statements in a fashion similar to the Java capture conversion \cite{WildFJ}. % %Type inference adds \texttt{let} statements in a fashion similar to the Java capture conversion \cite{WildFJ}.
%We wrap every parameter of a method invocation in \texttt{let} statement unknowing if a capture conversion is necessary. % %We wrap every parameter of a method invocation in \texttt{let} statement unknowing if a capture conversion is necessary.
Figure \ref{fig:tletexpr} shows type rules for fields and method calls. % Figure \ref{fig:tletexpr} shows type rules for fields and method calls.
They have been merged with let statements and simplified. % They have been merged with let statements and simplified.
The let statements and the type $\wcNtype{\Delta'}{N}$, which the type inference algorithm can freely choose, % The let statements and the type $\wcNtype{\Delta'}{N}$, which the type inference algorithm can freely choose,
are necessary for the soundness proof. % are necessary for the soundness proof.
%TODO: Show that well-formed implies witnessed! % %TODO: Show that well-formed implies witnessed!
We change the type rules to require the well-formedness instead of the witnessed property. % We change the type rules to require the well-formedness instead of the witnessed property.
See figure \ref{fig:well-formedness}. % See figure \ref{fig:well-formedness}.
Our well-formedness criteria is more restrictive than the ones used for \wildFJ{}. % Our well-formedness criteria is more restrictive than the ones used for \wildFJ{}.
\cite{WildcardsNeedWitnessProtection} works with the two different judgements $\ok$ and \texttt{witnessed}. % \cite{WildcardsNeedWitnessProtection} works with the two different judgements $\ok$ and \texttt{witnessed}.
With \texttt{witnessed} being the stronger one. % With \texttt{witnessed} being the stronger one.
We rephrased the $\ok$ judgement to include \texttt{witnessed} aswell. % We rephrased the $\ok$ judgement to include \texttt{witnessed} aswell.
$\type{T} \ \ok$ in this paper means $\type{T} \ \ok$ and $\type{T} \ \texttt{witnessed}$ in the sense of the \wildFJ{} type rules. % $\type{T} \ \ok$ in this paper means $\type{T} \ \ok$ and $\type{T} \ \texttt{witnessed}$ in the sense of the \wildFJ{} type rules.
Java's type system is complex enough as it is. Simplification, when possible, is always appreciated. % Java's type system is complex enough as it is. Simplification, when possible, is always appreciated.
Our $\ok$ rule may not be able to express every corner of the Java type system, but is sufficient to show soundness regarding type inference for a Java core calculus. % Our $\ok$ rule may not be able to express every corner of the Java type system, but is sufficient to show soundness regarding type inference for a Java core calculus.
The \rulename{WF-Class} rule requires upper and lower bounds to be direct subtypes of each other $\type{L} <: \type{U}$. % The \rulename{WF-Class} rule requires upper and lower bounds to be direct subtypes of each other $\type{L} <: \type{U}$.
The type rules from \cite{WildcardsNeedWitnessProtection} use a witness type instead. % The type rules from \cite{WildcardsNeedWitnessProtection} use a witness type instead.
Stating that a type is well formed (\texttt{witnessed}) if there exists atleast one simple type $\type{N}$ as a possible subtype: % Stating that a type is well formed (\texttt{witnessed}) if there exists atleast one simple type $\type{N}$ as a possible subtype:
$\Delta \vdash \type{N} <: \wctype{\Delta'}{C}{\ol{T}}$. % $\Delta \vdash \type{N} <: \wctype{\Delta'}{C}{\ol{T}}$.
A witness type is easy to find by replacing every wildcard $\ol{W}$ in $\wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$ by its upper bound: % A witness type is easy to find by replacing every wildcard $\ol{W}$ in $\wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$ by its upper bound:
$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} <: \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$. % $\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} <: \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$.
$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} \ \texttt{witnessed}$ is given due to $\Delta \vdash \ol{T}, \ol{L}, \ol{U} \ \ok$ % $\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} \ \texttt{witnessed}$ is given due to $\Delta \vdash \ol{T}, \ol{L}, \ol{U} \ \ok$
and $\Delta \vdash \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}} \ \ok$. % and $\Delta \vdash \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}} \ \ok$.
\begin{figure}[tp] % \begin{figure}[tp]
$\begin{array}{l} % $\begin{array}{l}
\typerule{T-Var}\\ % \typerule{T-Var}\\
\begin{array}{@{}c} % \begin{array}{@{}c}
x : \type{T} \in \Gamma % x : \type{T} \in \Gamma
\\ % \\
\hline % \hline
\vspace*{-0.3cm}\\ % \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash x : \type{T} % \Delta | \Gamma \vdash x : \type{T}
\end{array} % \end{array}
\end{array}$ % \end{array}$
\\[1em] % \\[1em]
$\begin{array}{l} % $\begin{array}{l}
\typerule{T-New}\\ % \typerule{T-New}\\
\begin{array}{@{}c} % \begin{array}{@{}c}
\Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} \ \ok \quad \quad % \Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} \ \ok \quad \quad
\text{fields}(\exptype{C}{\ol{T}}) = \overline{\type{U}\ f} \quad \quad % \text{fields}(\exptype{C}{\ol{T}}) = \overline{\type{U}\ f} \quad \quad
\Delta | \Gamma \vdash \overline{t : \type{S}} \quad \quad % \Delta | \Gamma \vdash \overline{t : \type{S}} \quad \quad
\Delta \vdash \overline{\type{S}} <: \overline{\wcNtype{\Delta}{N}} \\ % \Delta \vdash \overline{\type{S}} <: \overline{\wcNtype{\Delta}{N}} \\
\Delta, \overline{\Delta} \vdash \overline{\type{N}} <: \overline{\type{U}} \quad \quad % \Delta, \overline{\Delta} \vdash \overline{\type{N}} <: \overline{\type{U}} \quad \quad
\Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} <: \type{T} \quad \quad % \Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} <: \type{T} \quad \quad
\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} \quad \quad % \overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} \quad \quad
\Delta \vdash \type{T}, \overline{\wcNtype{\Delta}{N}} \ \ok % \Delta \vdash \type{T}, \overline{\wcNtype{\Delta}{N}} \ \ok
\\ % \\
\hline % \hline
\vspace*{-0.3cm}\\ % \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \letstmt{\ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}{\texttt{new} \ \exptype{C}{\ol{T}}(\overline{t})} : \type{T} % \Delta | \Gamma \vdash \letstmt{\ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}{\texttt{new} \ \exptype{C}{\ol{T}}(\overline{t})} : \type{T}
\end{array} % \end{array}
\end{array}$ % \end{array}$
\\[1em] % \\[1em]
$\begin{array}{l} % $\begin{array}{l}
\typerule{T-Field}\\ % \typerule{T-Field}\\
\begin{array}{@{}c} % \begin{array}{@{}c}
\Delta | \Gamma \vdash \texttt{t} : \type{T} \quad \quad % \Delta | \Gamma \vdash \texttt{t} : \type{T} \quad \quad
\Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad % \Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
\textit{fields}(\type{N}) = \ol{U\ f} \\ % \textit{fields}(\type{N}) = \ol{U\ f} \\
\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad % \Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad % \text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok % \Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
\\ % \\
\hline % \hline
\vspace*{-0.3cm}\\ % \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \texttt{let}\ x : \wcNtype{\Delta'}{N} = \texttt{t} \ \texttt{in}\ x.\texttt{f}_i : \type{S} % \Delta | \Gamma \vdash \texttt{let}\ x : \wcNtype{\Delta'}{N} = \texttt{t} \ \texttt{in}\ x.\texttt{f}_i : \type{S}
\end{array} % \end{array}
\end{array}$ % \end{array}$
\\[1em] % \\[1em]
$\begin{array}{l} % $\begin{array}{l}
\typerule{T-Call}\\ % \typerule{T-Call}\\
\begin{array}{@{}c} % \begin{array}{@{}c}
\Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad % \Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad
\textit{mtype}(\texttt{m}, \type{N}) = \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \quad \quad % \textit{mtype}(\texttt{m}, \type{N}) = \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \quad \quad
\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'} % \Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}
\\ % \\
\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad % \Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad
\Delta | \Gamma \vdash \texttt{t}_r : \type{T}_r \quad \quad % \Delta | \Gamma \vdash \texttt{t}_r : \type{T}_r \quad \quad
\Delta | \Gamma \vdash \ol{t} : \ol{T} \quad \quad % \Delta | \Gamma \vdash \ol{t} : \ol{T} \quad \quad
\Delta \vdash \type{T}_r <: \wcNtype{\Delta'}{N} \quad \quad % \Delta \vdash \type{T}_r <: \wcNtype{\Delta'}{N} \quad \quad
\Delta \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}} % \Delta \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}
\\ % \\
\Delta \vdash \type{T}, \wcNtype{\Delta'}{N}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad % \Delta \vdash \type{T}, \wcNtype{\Delta'}{N}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad
\Delta, \Delta', \Delta'' \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad % \Delta, \Delta', \Delta'' \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad % \text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} % \overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})}
\\ % \\
\hline % \hline
\vspace*{-0.3cm}\\ % \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \letstmt{x : \wcNtype{\Delta'}{N} = t_r, \ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}} % \Delta | \Gamma \vdash \letstmt{x : \wcNtype{\Delta'}{N} = t_r, \ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}
{\texttt{x}.\generics{\ol{S}}\texttt{m}(\ol{x})} : \type{T} % {\texttt{x}.\generics{\ol{S}}\texttt{m}(\ol{x})} : \type{T}
\end{array} % \end{array}
\end{array}$ % \end{array}$
\\[1em] % \\[1em]
$\begin{array}{l} % $\begin{array}{l}
\typerule{T-Elvis}\\ % \typerule{T-Elvis}\\
\begin{array}{@{}c} % \begin{array}{@{}c}
\Delta | \Gamma \vdash t_1 : \type{T}_1 \quad \quad % \Delta | \Gamma \vdash t_1 : \type{T}_1 \quad \quad
\Delta | \Gamma \vdash t_2 : \type{T}_2 \quad \quad % \Delta | \Gamma \vdash t_2 : \type{T}_2 \quad \quad
\Delta \vdash \type{T}_1 <: \type{T} \quad \quad % \Delta \vdash \type{T}_1 <: \type{T} \quad \quad
\Delta \vdash \type{T}_2 <: \type{T} % \Delta \vdash \type{T}_2 <: \type{T}
\\ % \\
\hline % \hline
\vspace*{-0.3cm}\\ % \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash t_1 \elvis{} t_2 : \type{T} % \Delta | \Gamma \vdash t_1 \elvis{} t_2 : \type{T}
\end{array} % \end{array}
\end{array}$ % \end{array}$
\\[1em] % \\[1em]
$\begin{array}{l} % $\begin{array}{l}
\typerule{T-Method}\\ % \typerule{T-Method}\\
\begin{array}{@{}c} % \begin{array}{@{}c}
\text{dom}(\Delta)=\ol{X} \quad \quad % \text{dom}(\Delta)=\ol{X} \quad \quad
\Delta' = \overline{\type{Y} : \bot .. \type{U}} \quad \quad % \Delta' = \overline{\type{Y} : \bot .. \type{U}} \quad \quad
\Delta, \Delta' \vdash \ol{U}, \type{T}, \ol{T}\ \ok \quad \quad % \Delta, \Delta' \vdash \ol{U}, \type{T}, \ol{T}\ \ok \quad \quad
\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ldots} \\ % \texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ldots} \\
\Delta, \Delta' | \overline{x:\type{T}}, \texttt{this} : \exptype{C}{\ol{X}} \vdash t:\type{S} \quad \quad % \Delta, \Delta' | \overline{x:\type{T}}, \texttt{this} : \exptype{C}{\ol{X}} \vdash t:\type{S} \quad \quad
\Delta, \Delta' \vdash \type{S} <: \type{T} \quad \quad % \Delta, \Delta' \vdash \type{S} <: \type{T} \quad \quad
\text{override}(\texttt{m}, \type{N}, \generics{\ol{Y \triangleleft U}}\ol{T} \to \type{T}) % \text{override}(\texttt{m}, \type{N}, \generics{\ol{Y \triangleleft U}}\ol{T} \to \type{T})
\\ % \\
\hline % \hline
\vspace*{-0.3cm}\\ % \vspace*{-0.3cm}\\
\Delta \vdash \generics{\ol{Y \triangleleft U}} \type{T} \ \texttt{m}(\overline{\type{T} \ x}) = t \ \ok \ \texttt{in C} % \Delta \vdash \generics{\ol{Y \triangleleft U}} \type{T} \ \texttt{m}(\overline{\type{T} \ x}) = t \ \ok \ \texttt{in C}
\end{array} % \end{array}
\end{array}$ % \end{array}$
\\[1em] % \\[1em]
$\begin{array}{l} % $\begin{array}{l}
\typerule{T-Class}\\ % \typerule{T-Class}\\
\begin{array}{@{}c} % \begin{array}{@{}c}
\Delta = \overline{\type{X} : \bot .. \type{U}} \quad \quad % \Delta = \overline{\type{X} : \bot .. \type{U}} \quad \quad
\Delta \vdash \ol{U}, \ol{T} \ \ok \quad \quad % \Delta \vdash \ol{U}, \ol{T} \ \ok \quad \quad
\Delta \vdash \type{N} \ \ok \quad \quad % \Delta \vdash \type{N} \ \ok \quad \quad
\Delta \vdash \ol{M} \ \ok \texttt{ in C} % \Delta \vdash \ol{M} \ \ok \texttt{ in C}
\\ % \\
\hline % \hline
\vspace*{-0.3cm}\\ % \vspace*{-0.3cm}\\
\texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M} } \ \ok % \texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M} } \ \ok
\end{array} % \end{array}
\end{array}$ % \end{array}$
\caption{T-Call and T-Field} \label{fig:tletexpr} % \caption{T-Call and T-Field} \label{fig:tletexpr}
\end{figure} % \end{figure}

3
tRules.tex
View File

@ -61,6 +61,9 @@ class List<A extends Object> {
as we consider them as data declarations which are given by the programmer. as we consider them as data declarations which are given by the programmer.
% They are inferred in for example \cite{plue14_3b} % They are inferred in for example \cite{plue14_3b}
\item We add the elvis operator ($\elvis{}$) to the syntax to easily showcase applications involving wildcard types. \item We add the elvis operator ($\elvis{}$) to the syntax to easily showcase applications involving wildcard types.
This operator is used to emulate Java's \texttt{if-else} expression but without the condition clause.
Our operator just returns one of the two given statements at random.
The point is that the return type of the elvis operator has to be a supertype of its two subexpressions.
\item The \texttt{new} expression does not require generic parameters. \item The \texttt{new} expression does not require generic parameters.
\item Every method has an unique name. \item Every method has an unique name.
The calculus does not include method overriding for simplicity. The calculus does not include method overriding for simplicity.

414
typeinference.tex Normal file
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@ -0,0 +1,414 @@
\section{Global Type Inference Algorithm}
\label{sec:glob-type-infer}
This section gives an overview of the global type inference algorithm
with examples and identifies the challenges that we had to overcome in
the design of the algorithm.
\begin{figure}[h]
\begin{minipage}{0.49\textwidth}
\begin{lstlisting}[style=tfgj, caption=Valid Java program, label=lst:addExample]
<A> List<A> add(List<A> l, A v)
List<? super String> l = ...;
add(l, "String");
\end{lstlisting}
\end{minipage}\hfill
\begin{minipage}{0.49\textwidth}
\begin{lstlisting}[style=tamedfj, caption=\TamedFJ{} representation, label=lst:addExampleLet]
<A> List<A> add(List<A> l, A v)
List<? super String> l = ...;
let v:(*@$\tv{v}$@*) = l
in add(v, "String");
\end{lstlisting}
\end{minipage}\\
\begin{minipage}{0.49\textwidth}
\begin{lstlisting}[style=constraints, caption=Constraints, label=lst:addExampleCons]
(*@$\wctype{\wildcard{X}{\type{String}}{\bot}}{List}{\rwildcard{X}} \lessdot \tv{v}$@*)
(*@$\tv{v} \lessdotCC \exptype{List}{\wtv{a}}$@*)
(*@$\type{String} \lessdot \wtv{a}$@*)
\end{lstlisting}
\end{minipage}\hfill
\begin{minipage}{0.49\textwidth}
\begin{lstlisting}[style=tamedfj, caption=Type solution, label=lst:addExampleSolution]
<A> List<A> add(List<A> l, A v)
List<? super String> l = ...;
let l2:(*@$\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}}$@*) = l
in <X>add(l2, "String");
\end{lstlisting}
\end{minipage}
\end{figure}
% \begin{description}
% \item[input] \tifj{} program
% \item[output] type solution
% \item[postcondition] the type solution applied to the input must yield a valid \letfj{} program
% \end{description}
%Our algorithm is an extension of the \emph{Global Type Inference for Featherweight Generic Java}\cite{TIforFGJ} algorithm.
Listings~\ref{lst:addExample}, \ref{lst:addExampleLet}, \ref{lst:addExampleCons}, and
\ref{lst:addExampleSolution} showcase the global type inference algorithm step by step.
Note that regular Java code snippets are displayed in a grey box, \TamedFJ{} programs are yellow
and constraints have a red background.
%This example is fully typed except for the method call \texttt{add} missing its type parameters.
In the Java code in Listing~\ref{lst:addExample}, the type of variable
\texttt{l} is an existential type so that it has to undergo a capture conversion
before being passed to a method call.
To this end we convert the program to A-Normal form (Listing~\ref{lst:addExampleLet}),
which introduces a let statement defining a new variable \texttt{v}.
The constraint generation step also assigns type placeholders to all variables missing a type annotation.
%In this case the variable \texttt{v} gets the type placeholder $\tv{v}$ (also shown in listing \ref{lst:addExampleCons})
%during the constraint generation step.
In this case the algorithm puts the type
placeholder $\tv{v}$ for the type of \texttt{v} before generating constraints (see Listing~\ref{lst:addExampleCons}).
These constraints mirror the typing rules of the \TamedFJ{} calculus (section~\ref{sec:tifj}).
% During the constraint generation step the type of the variable \texttt{v} is unknown
% and given the type placeholder $\tv{v}$.
The call to \texttt{add} generates the \emph{capture constraint} $\tv{v} \lessdotCC \exptype{List}{\wtv{a}}$.
This constraint ($\lessdotCC$) is a kind of subtype constraint, which additionally
expresses that the left side of the constraint is subject to a capture conversion.
Next, the unification algorithm \unify{} (section~\ref{sec:unify}) solves the constraints.
Havin the constraints in listing \ref{lst:addExampleCons} as an input \unify{} changes the constraint
$\wctype{\wildcard{X}{\type{String}}{\bot}}{List}{\rwildcard{X}} \lessdot \tv{v}$
to $\tv{v} \doteq \wctype{\wildcard{X}{\type{String}}{\bot}}{List}{\rwildcard{X}}$
therby finding a type solution for $\tv{v}$.
Afterwards every occurence of $\tv{v}$ in the constraint set is replaced by $\wctype{\wildcard{X}{\type{String}}{\bot}}{List}{\rwildcard{X}}$
leaving us with the following constraints which are now subject to a capture conversion by the \unify{} algorithm:
% Initially, no type is assigned to $\tv{v}$.
% % During the course of \unify{} more and more types emerge.
% As soon as a concrete type for $\tv{v}$ is appears during constraint
% solving, \unify{} can conduct a capture conversion if needed.
% %The constraints where this is possible are marked as capture constraints.
% In this example, $\tv{v}$ will be set to
% $\wctype{\wildcard{X}{\type{String}}{\bot}}{List}{\rwildcard{X}}$
% leaving us with the following constraints:
\begin{displaymath}
\prftree[r]{Capture}{
\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}, \, \type{String} \lessdotCC \wtv{a}
}{
\wildcard{Y}{\type{Object}}{\type{String}} \wcSep \exptype{List}{\rwildcard{Y}} \lessdot \exptype{List}{\wtv{a}}, \, \type{String} \lessdot \wtv{a}
}
\end{displaymath}
%Capture Constraints $(\wctype{\rwildcard{X}}{C}{\rwildcard{X}} \lessdotCC \type{T})$ allow for a capture conversion,
%which converts a constraint of the form $(\wctype{\rwildcard{X}}{C}{\rwildcard{X}} \lessdotCC \type{T})$ to $(\exptype{C}{\rwildcard{X}} \lessdot \type{T})$
The constraint $\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}$
allows \unify{} to do a capture conversion to $\exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$.
The captured wildcard $\rwildcard{X}$ gets a fresh name \texttt{Y}, which is stored in the wildcard environment of the \unify{} algorithm.
Substituting \texttt{Y} for $\wtv{a}$ yields the constraints almost
solved: $\exptype{List}{\rwildcard{Y}} \lessdot
\exptype{List}{\rwildcard{Y}}$, $\type{String} \lessdot
\rwildcard{Y}$.
The first constraint is obviously satisfied and $\type{String} \lessdot \rwildcard{Y}$ is satisfied
because $\rwildcard{Y}$ has $\type{String}$ as lower bound.
A correct Featherweight Java program including all type annotations and an explicit capture conversion via let statement is shown in Listing \ref{lst:addExampleSolution}.
This program can be deduced from the solution of the \unify{} algorithm.% presented in Section~\ref{sec:unify}.
In the body of the let statement the type $\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}}$
becomes $\exptype{List}{\rwildcard{X}}$ and the capture converted wildcard $\wildcard{X}{\type{Object}}{\type{String}}$ can be used as
a type parameter of the call \texttt{<X>add(v, "String")}.
% The input to our type inference algorithm is a modified version of the calculus in \cite{WildcardsNeedWitnessProtection} (see chapter \ref{sec:tifj}).
% First \fjtype{} (see section \ref{chapter:constraintGeneration}) generates constraints
% and afterwards \unify{} (section \ref{sec:unify}) computes a solution for the given constraint set.
% Constraints consist out of subtype constraints $(\type{T} \lessdot \type{T})$ and capture constraints $(\type{T} \lessdotCC \type{T})$.
% \textit{Note:} a type $\type{T}$ can either be a named type, a type placeholder or a wildcard type placeholder.
% A subtype constraint is satisfied if the left side is a subtype of the right side according to the rules in figure \ref{fig:subtyping}.
% \textit{Example:} $\exptype{List}{\ntv{a}} \lessdot \exptype{List}{\type{String}}$ is fulfilled by replacing type placeholder $\ntv{a}$ with the type $\type{String}$.
% Subtype constraints and type placeholders act the same as the ones used in \emph{Type Inference for Featherweight Generic Java} \cite{TIforFGJ}.
% The novel capture constraints and wildcard placeholders are needed for method invocations involving wildcards.
% The central piece of this type inference algorithm, the \unify{} process, is described with implication rules (chapter \ref{sec:unify}).
% We try to keep the branching at a minimal amount to improve runtime behavior.
% Also the transformation steps of the \unify{} algorithm are directly related to the subtyping rules of our calculus.
% There are no informal parts in our \unify{} algorithm.
% It solely consist out of transformation rules which are bound to simple checks.
\subsection{Notes on Capture Constraints}
Capture constraints must be stored in a multiset so that it is possible that two syntactically equal constraints remain in the same set.
The equality relation on Capture constraints is not reflexive.
Capture Constraints are bound to a variable.
For example a let statement like \lstinline{let x = v in x.get()} will create the capture constraint
$\tv{x} \lessdotCC_x \exptype{List}{\wtv{a}}$.
This time we annotated the capture constraint with an $x$ to show its relation to the variable \texttt{x}.
Let's do the same with the constraints generated by the \texttt{concat} method invocation in listing \ref{lst:faultyConcat},
creating the constraints \ref{lst:sameConstraints}.
\begin{figure}
\begin{minipage}[t]{0.49\textwidth}
\begin{lstlisting}[caption=Faulty Method Call,label=lst:faultyConcat,style=TamedFJ]{tamedfj}
List<?> v = ...;
let x = v in
let y = v in
concat(x, y) // Error!
\end{lstlisting}\end{minipage}
\hfill
\begin{minipage}[t]{0.49\textwidth}
\begin{lstlisting}[caption=Annotated constraints,mathescape=true,style=constraints,label=lst:sameConstraints]
$\tv{x} \lessdotCC_\texttt{x} \exptype{List}{\wtv{a}}, \wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdot \tv{x}$
$\tv{y} \lessdotCC_\texttt{y} \exptype{List}{\wtv{a}}, \wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdot \tv{y}$
\end{lstlisting}
\end{minipage}
\end{figure}
During the \unify{} process it could happen that two syntactically equal capture constraints evolve,
but they are not the same because they are each linked to a different let introduced variable.
In this example this happens when we substitute $\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ for $\tv{x}$ and $\tv{y}$
resulting in:
%For example by substituting $[\wctype{\rwildcard{X}}{List}{\rwildcard{X}}/\tv{x}]$ and $[\wctype{\rwildcard{X}}{List}{\rwildcard{X}}/\tv{y}]$:
\begin{displaymath}
\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC_x \exptype{List}{\wtv{a}}, \wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC_y \exptype{List}{\wtv{a}}
\end{displaymath}
Thanks to the original annotations we can still see that those are different constraints.
After \unify{} uses the \rulename{Capture} rule on those constraints
it gets obvious that this constraint set is unsolvable:
\begin{displaymath}
\exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}},
\exptype{List}{\rwildcard{Y}} \lessdot \exptype{List}{\wtv{a}}
\end{displaymath}
%In this paper we do not annotate capture constraint with their source let statement.
This paper will not annotate capture constraints with variable names.
Instead we consider every capture constraint as distinct to other capture constraints even when syntactically the same,
because we know that each of them originates from a different let statement.
\textit{Hint:} An implementation of this algorithm has to consider that seemingly equal capture constraints are actually not the same
and has to allow doubles in the constraint set.
% %We see the equality relation on Capture constraints is not reflexive.
% A capture constraint is never equal to another capture constraint even when structurally the same
% ($\type{T} \lessdotCC \type{S} \neq \type{T} \lessdotCC \type{S}$).
% This is necessary to solve challenge \ref{challenge:1}.
% A capture constraint is bound to a specific let statement.
\textit{Note:}
In the special case \lstinline{let x = v in concat(x,x)} the constraints would look like
$\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC_\texttt{x} \exptype{List}{\wtv{a}},
\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC_\texttt{x} \exptype{List}{\wtv{a}}$
and we could actually delete one of them without loosing information.
But this case will never occur in our algorithm, because the let
statements for our input programs are generated by transformation to ANF (see \ref{sec:anfTransformation}).
\subsection{Challenges}\label{challenges}
%TODO: Wildcard subtyping is infinite see \cite{TamingWildcards}
% Wildcards are not reflexive.
% ( on the equals property ), every wildcard has to be capture converted when leaving its scope
% do not substitute free type variables
Global type inference for Featherweight Generic Java without wildcards
has been considered elsewher \cite{TIforFGJ}.
Adding wildcards to the calculus created new problems, which have not
been recognized by existing work on type unification for Java with wildcards~\cite{plue09_1}.
% what is the problem?
% Java does invisible capture conversion steps
Java's wildcard types are represented as existential types that have to be opened before they can be used.
Opening can either be done implicitly
(\cite{aModelForJavaWithWildcards}, \cite{javaTIisBroken}) or
explicitly via a let statement (\cite{WildcardsNeedWitnessProtection}).
For all variations it is vital to know the argument types with which a method is called.
In the absence of type annotations,
we do not know where an existential type will emerge and where a capture conversion is necessary.
Rather, the type inference algorithm has to figure out the placement
of existentials.
We identified three main challenges related to Java wildcards and global type inference.
\begin{enumerate}
\item \label{challenge:1}
One challenge is to design the algorithm in a way that it finds a
correct solution for programs like the one shown in Listing~\ref{lst:addExample}
and rejects programs like the one in Listing~\ref{lst:concatError}.
The first one is a valid Java program,
because the type \texttt{List<? super String>} is \textit{capture converted} to a fresh type variable $\rwildcard{X}$
which is used as the generic method parameter for the call to \texttt{add} as shown in Listing~\ref{lst:addExampleLet}.
Because we know that the type \texttt{String} is a subtype of the free variable $\rwildcard{X}$,
it is safe to pass \texttt{"String"} for the first parameter of the function.
The second program shown in Listing~\ref{lst:concatError} is incorrect.
The method call to \texttt{concat} with two wildcard lists is unsound.
The element types of the lists may be unrelated, therefore the call to \texttt{concat} cannot succeed.
The problem gets apparent if we try to write the \texttt{concat}
method call in the \TamedFJ{} calculus
(Listing~\ref{lst:concatTamedFJ}):
\texttt{l1'} and \texttt{l2'} are two different lists inside the body of the let statements, namely
$\exptype{List}{\rwildcard{X}}$ and $\exptype{List}{\rwildcard{Y}}$.
For the method call \texttt{concat(x1, x2)} no replacement for the generic \texttt{A}
exists to satisfy
$\exptype{List}{\type{A}} <: \exptype{List}{\type{X}},
\exptype{List}{\type{A}} <: \exptype{List}{\type{Y}}$.
% \textbf{Solution:}
% Capture Conversion during Unify.
% \item
% \unify{} transforms a constraint set into a correct type solution by
% gradually assigning types to type placeholders during that process.
% Solved constraints are removed and never considered again.
% In the following example, \unify{} solves the constraint generated by the expression
% \texttt{l.add(l.head())} first, which results in $\ntv{l} \lessdot \exptype{List}{\wtv{a}}$.
% \begin{verbatim}
% anyList() = new List<String>() :? new List<Integer>()
% add(anyList(), anyList().head());
% \end{verbatim}
% The type for \texttt{l} can be an arbitrary list, but it has to be a invariant one.
% Assigning a \texttt{List<?>} for \texttt{l} is unsound, because the type list hiding behind
% \texttt{List<?>} could be a different one for the \texttt{add} call than the \texttt{head} method call.
% An additional constraint $\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$
% is solved by removing the wildcard $\rwildcard{X}$ if possible.
% This problem is solved by ANF transformation
\item
% This problem is solved by assuming everything is a wildcard and lateron erasing excessive wildcards
% this is solved by having wildcards bound to a type. But this makes it necessary to remove wildcards lateron otherwise Unify would have to backtrack
The program in Listing~\ref{shuffleExample} exhibits a challenge involving wildcards and subtyping.
The method call \texttt{shuffle(l)} is incorrect, because \texttt{l} has type
$\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$ representing a list of unknown lists.
However, $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$ is a subtype of
$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$ which represents a list of lists, all of the same type $\rwildcard{X}$,
and can be passed safely to \texttt{shuffle}.
This behavior can also be explained by looking at the types and their capture converted versions:
\begin{center}
\begin{tabular}{l | l | l}
Java type & \TamedFJ{} representation & Capture Conversion \\
\hline
$\exptype{List}{\exptype{List}{\texttt{?}}}$ & $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$ & $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$ \\
$\exptype{List2D}{\texttt{?}}$ & $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$ & $\exptype{List2D}{\rwildcard{X}}$ \\
%Supertype of $\exptype{List2D}{\texttt{?}}$ & $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$ & $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$ \\
\end{tabular}
\end{center}
%The direct supertype of $\exptype{List2D}{\rwildcard{X}}$ is $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$.
%One problem is the divergence between denotable and expressable types in Java \cite{semanticWildcardModel}.
%A wildcard in the Java syntax has no name and is bound to its enclosing type:
%$\exptype{List}{\exptype{List}{\type{?}}}$ equates to $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$.
%During type checking \emph{intermediate types}
%can emerge, which have no equivalent in the Java syntax.
\begin{lstlisting}[style=java,label=shuffleExample,caption=Intermediate Types Example]
class List<X> extends Object {...}
class List2D<X> extends List<List<X>> {...}
<X> void shuffle(List<List<X>> list) {...}
List<List<?>> l = ...;
List2D<?> l2d = ...;
shuffle(l); // Error
shuffle(l2d); // Valid
\end{lstlisting}
% Java is using local type inference to allow method invocations which are not describable with regular Java types.
% The \texttt{shuffle} method in this case is invoked with the type $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$
% which is a subtype of $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$.
% After capture conversion \texttt{l2d'} has the type $\exptype{List}{\exptype{List}{\rwildcard{X}}}$
% and \texttt{shuffle} can be invoked with the type parameter $\rwildcard{X}$:
% \begin{lstlisting}[style=TamedFJ]
% let l2d' : (*@$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$@*) = l2d in <X>shuffle(l2d')
% \end{lstlisting}
% For the example shown in listing \ref{shuffleExample} the method call \texttt{shuffle(l2d)} creates the constraints:
% \begin{constraintset}
% \begin{center}
% $
% \begin{array}{l}
% \wctype{\rwildcard{X}}{List2D}{\rwildcard{X}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}
% \\
% \hline
% \wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}
% \\
% \hline
% \textit{Capture Conversion:}\
% \exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\wtv{x}}}
% \\
% \hline
% \textit{Solution:} \wtv{x} \doteq \rwildcard{X} \implies \exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\rwildcard{X}}}
% \end{array}
% $
% \end{center}
% \end{constraintset}
Given such a program the type inference algorithm has to allow the call \texttt{shuffle(l2d)} and
decline the call to \texttt{shuffle(l)}.
% The method call \texttt{shuffle(l)} is invalid however,
% because \texttt{l} has the type
% $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$.
% There is no solution for the subtype constraint:
% $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}$
\item \label{challenge3} \textbf{Free variables cannot leave their scope}:
Let's assume we have a variable \texttt{ls} with type $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$
%When calling the \texttt{id} function with an element of this list we have to apply capture conversion.
and the following program:
\noindent
\begin{minipage}{0.62\textwidth}
\begin{lstlisting}
let x : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) = ls.get(0) in id(x) : (*@$\ntv{z}$@*)
\end{lstlisting}\end{minipage}
\hfill
\begin{minipage}{0.36\textwidth}
\begin{lstlisting}[style=constraints]
(*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{x}}$@*),
(*@$\exptype{List}{\wtv{x}} \lessdot \ntv{z},$@*)
\end{lstlisting}
\end{minipage}
% the variable z must not contain free variables (TODO)
Take the Java program in listing \ref{lst:mapExample} for example.
It uses map to apply a polymorphic method \texttt{id} to every element of a list
$\expr{l} :
\exptype{List}{\wctype{\rwildcard{A}}{List}{\rwildcard{A}}}$.
How do we get the \unify{} algorithm to determine the correct type for the
variable \expr{l2}?
Although we do not specify constraint generation for language constructs like
lambda expressions used in this example,
we can imagine that the constraints have to look like in Listing~\ref{lst:mapExampleCons}.
\begin{minipage}{0.45\textwidth}
\begin{lstlisting}[caption=List Map Example,label=lst:mapExample]
<X> List<X> id(List<X> l){ ... }
List<List<?>> ls;
l2 = l.map(x -> id(x));
\end{lstlisting}\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
\begin{lstlisting}[style=constraints, caption=Constraints, label=lst:mapExampleCons]
(*@$\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\wtv{x}}$@*)
(*@$\exptype{List}{\wtv{x}} \lessdot \tv{z},$@*)
(*@$\exptype{List}{\tv{z}} \lessdot \tv{l2}$@*)
\end{lstlisting}
\end{minipage}
The constraints
$\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\wtv{x}},
\exptype{List}{\wtv{x}} \lessdot \tv{z}$
stem from the body of the lambda expression
\texttt{id(x)}.
\textit{For clarification:} This method call would be represented as the following expression in \letfj{}:
\texttt{let x1 :$\wctype{\rwildcard{A}}{List}{\rwildcard{A}}$ = x in id(x) :$\tv{z}$}
The T-Let rule prevents us from using free variables created by the method call to \expr{id}
to be used in the return type $\tv{z}$.
But this restriction has to be communicated to the \unify{} algorithm,
which does not know about the origin and context of
the constraints.
If we naively substitute $\sigma(\tv{z}) = \exptype{List}{\rwildcard{A}}$
the return type of the \texttt{map} function would be the type
$\exptype{List}{\exptype{List}{\rwildcard{A}}}$, which would be unsound.
The type of \expr{l2} is the same as the one of \expr{l}:
$\exptype{List}{\wctype{\rwildcard{A}}{List}{\rwildcard{A}}}$
\textbf{Solution:}
We solve this issue by distinguishing between wildcard placeholders
and normal placeholders and introducing them as needed.
$\ntv{z}$ is a normal placeholder and is not allowed to contain free variables.
\end{enumerate}

86
unify.tex
View File

@ -680,16 +680,13 @@ exists the wildcard $\rwildcard{X}$ has to be removed by setting both lower and
% } % }
% \end{lstlisting} % \end{lstlisting}
Take the introduction example from listing \ref{lst:intro-example}. Let's have a look at the constraints generated by the introduction example in listing \ref{lst:intro-example-typeless}:
Constraints for the untyped \texttt{genList} method: \begin{lstlisting}[style=constraints]
\begin{constraintset} (*@$\exptype{List}{\ntv{x}} \lessdot {\ntv{r}}, \type{String} \lessdot {\ntv{x}}$@*),
$\begin{array}{l} (*@$\exptype{List}{\ntv{y}} \lessdot {\ntv{r}}, \type{Integer} \lessdot {\ntv{y}}$@*)
\exptype{List}{\ntv{x}} \lessdot {\ntv{r}}, \type{String} \lessdot {\ntv{x}}, \end{lstlisting}
\exptype{List}{\ntv{y}} \lessdot {\ntv{r}}, \type{Integer} \lessdot {\ntv{y}}
\end{array}$
\end{constraintset}
\unify{} executes the following steps to generate a solved constraint set:
\begin{displaymath} \begin{displaymath}
\prftree[r]{\rulename{Subst}} \prftree[r]{\rulename{Subst}}
{ {
@ -786,55 +783,22 @@ The \texttt{concat} method takes two lists of the same generic type.
% Free variables are not allowed to leave their scope. % Free variables are not allowed to leave their scope.
% This is ensured by type variables which are not allowed to be assigned type holding free variables. % This is ensured by type variables which are not allowed to be assigned type holding free variables.
\subsection{Algorithm} % \subsection{Algorithm}
With \texttt{C} being class names and \texttt{A} being wildcard names. % With \texttt{C} being class names and \texttt{A} being wildcard names.
The wildcard type $\wildcard{X}{U}{L}$ consist of an upper bound $\type{U}$, a lower bound $\type{L}$ % The wildcard type $\wildcard{X}{U}{L}$ consist of an upper bound $\type{U}$, a lower bound $\type{L}$
and a name $\mathtt{X}$. % and a name $\mathtt{X}$.
The \rulename{Tame} rule eliminates wildcards. %TODO % The \rulename{Tame} rule eliminates wildcards. %TODO
This is done by setting the upper and lower bound to the same value. % This is done by setting the upper and lower bound to the same value.
The \rulename{match} rule generates fresh wildcards $\overline{\wildcard{A}{\tv{u}}{\tv{l}}}$. % The \rulename{match} rule generates fresh wildcards $\overline{\wildcard{A}{\tv{u}}{\tv{l}}}$.
Their upper and lower bounds are fresh type variables. % Their upper and lower bounds are fresh type variables.
\unify{} is able to remove wildcards by assigning their upper and lower bounds the same type % \unify{} is able to remove wildcards by assigning their upper and lower bounds the same type
(Def: $\type{Object} = \wildcard{A}{Object}{Object}$ by definition \ref{def:equal}). % (Def: $\type{Object} = \wildcard{A}{Object}{Object}$ by definition \ref{def:equal}).
This is used by the \rulename{Tame} rule. % This is used by the \rulename{Tame} rule.
\textbf{Helper functions:}
\begin{description}
\item[$\tph{}$] returns all type placeholders inside a given type.
\textit{Example:} $\tph(\wctype{\wildcard{X}{\tv{a}}{\bot}}{Pair}{\wtv{b},\rwildcard{X}}) = \set{\tv{a}, \wtv{b}}$
%\textbf{Wildcard renaming}\\
\item[Wildcard renaming:]
The \rulename{Reduce} rule separates wildcards from their environment.
At this point each wildcard gets a new and unique name.
To only rename the respective wildcards the reduce rule renames wildcards up to alpha conversion:
($[\ol{C}/\ol{B}]$ in the \rulename{reduce} rule)
\begin{align*}
[\type{A}/\type{B}]\type{B} &= \type{A} \\
[\type{A}/\type{B}]\type{C} &= \type{C} & \text{if}\ \type{B} \neq \type{C}\\
[\type{A}/\type{B}]\wcNtype{\overline{\wildcard{X}{\type{U}}{\type{L}}}}{N} &= \wcNtype{\overline{\wildcard{X}{[\type{A}/\type{B}]\type{U}}{[\type{A}/\type{B}]\type{L}}}}{[\type{A}/\type{B}]N} & \text{if}\ \type{B} \notin \overline{\rwildcard{X}} \\
[\type{A}/\type{B}]\wcNtype{\overline{\wildcard{X}{\type{U}}{\type{L}}}}{N} &= \wcNtype{\overline{\wildcard{X}{\type{U}}{\type{L}}}}{N} & \text{if}\ \type{B} \in \overline{\rwildcard{X}} \\
\end{align*}
\item[Free Variables:]
The $\text{fv}$ function assumes every wildcard type variable to be a free variable aswell.
% TODO: describe a function which determines free variables? or do an example?
\begin{align*}
\text{fv}(\rwildcard{A}) &= \set{ \rwildcard{A} } \\
\text{fv}(\tv{a}) &= \emptyset \\
%\text{fv}(\wtv{a}) &= \set{\wtv{a}} \\
\text{fv}(\wctype{\Delta}{C}{\ol{T}}) &= \set{\text{fv}(\type{T}) \mid \type{T} \in \ol{T}} / \text{dom}(\Delta) \\
\end{align*}
\item[Fresh Wildcards:]
$\text{fresh}\ \overline{\wildcard{A}{\tv{u}}{\tv{l}}}$ generates fresh wildcards.
The new names $\ol{A}$ are fresh, aswell as the type variables $\ol{\tv{u}}$ and $\ol{\tv{l}}$,
which are used for the upper and lower bounds.
\end{description}
% \noindent % \noindent
% \textbf{Example: (reduce-rule)} % \textbf{Example: (reduce-rule)}
% %The \ruleReduceWC{} resembles the \texttt{S-Exists} subtyping rule. % %The \ruleReduceWC{} resembles the \texttt{S-Exists} subtyping rule.
@ -1219,7 +1183,7 @@ $\set{\tv{a} \doteq \type{N}} \in C$ with $\text{fv}(\type{N}) \cap \Delta_{in}
\label{fig:step2-rules2} \label{fig:step2-rules2}
\end{figure} \end{figure}
Figure \ref{fig:step2-rules2} are special transformations aimed at free variables defined in the input environment $\Delta_{in}$. Figure \ref{fig:step2-rules} are special transformations aimed at free variables defined in the input environment $\Delta_{in}$.
Usually \rulename{Upper} takes care of $\rwildcard{X} \lessdot \tv{a}$ constraints, Usually \rulename{Upper} takes care of $\rwildcard{X} \lessdot \tv{a}$ constraints,
but if $\rwildcard{X}$ is an element of $\Delta_{in}$ we can also treat it as a regular type. but if $\rwildcard{X}$ is an element of $\Delta_{in}$ we can also treat it as a regular type.
This leaves us with the two possibilities \rulename{Subst-X} and \rulename{Gen-X} which is the same as \rulename{Upper}. This leaves us with the two possibilities \rulename{Subst-X} and \rulename{Gen-X} which is the same as \rulename{Upper}.
@ -1317,11 +1281,9 @@ $
\label{fig:generation-rules} \label{fig:generation-rules}
\end{figure} \end{figure}
\subsection{Explanations}
%The type reduction is done by the rules in figure \ref{fig:reductionRules}
\subsection{Examples} \subsection{Examples}
\textit{Example} of the type reduction rules in figure \ref{fig:reductionRules} with the input \textit{Example} of the type reduction rules in figure \ref{fig:reductionRules} with the input
$\wctype{\rwildcard{X}}{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \rwildcard{X}} \lessdot \exptype{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \wtv{a}}$ $\wctype{\rwildcard{X}}{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \rwildcard{X}} \lessdot \exptype{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \wtv{a}}$
The first step is the \rulename{Capture} rule. The first step is the \rulename{Capture} rule.
@ -1462,12 +1424,12 @@ is called and $\tv{a}$ gets a type, then this type can be an existential type as
% ( is this complete? X,Y.Pair<X,Y> <c X.Pair<X,X>) % ( is this complete? X,Y.Pair<X,Y> <c X.Pair<X,X>)
\subsection{Implementation} % \subsection{Implementation}
%List this under implementation details % %List this under implementation details
Every constraint that is not in solved form and is not able to be processed by any of the rules accounts as a error constraint and renders the constraint set unsolvable % Every constraint that is not in solved form and is not able to be processed by any of the rules accounts as a error constraint and renders the constraint set unsolvable
The new constraint generated by the adopt rule may be eliminated by the match rule. % The new constraint generated by the adopt rule may be eliminated by the match rule.
The adopt rule still needs to be applied only once per constraint. % The adopt rule still needs to be applied only once per constraint.
% \textbf{Eliminating Wildcards} % \textbf{Eliminating Wildcards}