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introduction.tex
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@ -550,12 +550,31 @@ is solved by removing the wildcard $\rwildcard{X}$ if possible.
this problem is solved by ANF transformation this problem is solved by ANF transformation
\item \textbf{Wildcards as Existential Types:} \item
One problem is the divergence between denotable and expressable types in Java \cite{semanticWildcardModel}. % This problem is solved by assuming everything is a wildcard and lateron erasing excessive wildcards
A wildcard in the Java syntax has no name and is bound to its enclosing type: % 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
$\exptype{List}{\exptype{List}{\type{?}}}$ equates to $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$. The program in listing \ref{shuffleExample} shows a challenge involving wildcards and subtyping.
During type checking \emph{intermediate types} The method call \texttt{shuffle(l)} is incorrect, because \texttt{l} has the type
can emerge, which have no equivalent in the Java syntax. $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$ representing a list of unknown lists.
Whereas $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$ is a subtype of
$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$ representing a list of lists, all of the same type $\rwildcard{X}$,
and can savely be passed 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] \begin{lstlisting}[style=java,label=shuffleExample,caption=Intermediate Types Example]
class List<X> extends Object {...} class List<X> extends Object {...}
@ -569,45 +588,64 @@ List2D<?> l2d = ...;
shuffle(l); // Error shuffle(l); // Error
shuffle(l2d); // Valid shuffle(l2d); // Valid
\end{lstlisting} \end{lstlisting}
Java is using local type inference to allow method invocations which are not describable with regular Java types. % 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 $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$ % The \texttt{shuffle} method in this case is invoked with the type $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$
which is a subtype of $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$. % which is a subtype of $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$.
After capture conversion \texttt{l2d'} has the type $\exptype{List}{\exptype{List}{\rwildcard{X}}}$ % After capture conversion \texttt{l2d'} has the type $\exptype{List}{\exptype{List}{\rwildcard{X}}}$
and \texttt{shuffle} can be invoked with the type parameter $\rwildcard{X}$: % and \texttt{shuffle} can be invoked with the type parameter $\rwildcard{X}$:
\begin{lstlisting}[style=TamedFJ] % \begin{lstlisting}[style=TamedFJ]
let l2d' : (*@$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$@*) = l2d in <X>shuffle(l2d') % let l2d' : (*@$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$@*) = l2d in <X>shuffle(l2d')
\end{lstlisting} % \end{lstlisting}
For the example shown in listing \ref{shuffleExample} the method call \texttt{shuffle(l2d)} creates the constraints: % For the example shown in listing \ref{shuffleExample} the method call \texttt{shuffle(l2d)} creates the constraints:
\begin{constraintset} % \begin{constraintset}
\begin{center} % \begin{center}
$ % $
\begin{array}{l} % \begin{array}{l}
\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}} % \wctype{\rwildcard{X}}{List2D}{\rwildcard{X}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}
\\ % \\
\hline % \hline
\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}} % \wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}
\\ % \\
\hline % \hline
\textit{Capture Conversion:}\ % \textit{Capture Conversion:}\
\exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\wtv{x}}} % \exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\wtv{x}}}
\\ % \\
\hline % \hline
\textit{Solution:} \wtv{x} \doteq \rwildcard{X} \implies \exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\rwildcard{X}}} % \textit{Solution:} \wtv{x} \doteq \rwildcard{X} \implies \exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\rwildcard{X}}}
\end{array} % \end{array}
$ % $
\end{center} % \end{center}
\end{constraintset} % \end{constraintset}
The method call \texttt{shuffle(l)} is invalid however, Given such a program our type inference algorithm has to allow the call \texttt{shuffle(l2d)} and
because \texttt{l} has the type decline the call to \texttt{shuffle(l)}.
$\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$.
There is no solution for the subtype constraint: % The method call \texttt{shuffle(l)} is invalid however,
$\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}$ % 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 leaver their scope}: \item \label{challenge3} \textbf{Free Variables cannot leaver 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 input 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
\begin{example}
Take the Java program in listing \ref{lst:mapExample} for example. Take the Java program in listing \ref{lst:mapExample} for example.
It maps every element of a list It maps every element of a list
$\expr{l} : \exptype{List}{\wctype{\rwildcard{A}}{List}{\rwildcard{A}}}$ $\expr{l} : \exptype{List}{\wctype{\rwildcard{A}}{List}{\rwildcard{A}}}$
@ -655,8 +693,6 @@ $\exptype{List}{\wctype{\rwildcard{A}}{List}{\rwildcard{A}}}$
We solve this by distinguishing between wildcard placeholders and normal placeholders. We solve this by distinguishing between wildcard placeholders and normal placeholders.
$\ntv{z}$ is a normal placeholder and is not allowed to contain free variables. $\ntv{z}$ is a normal placeholder and is not allowed to contain free variables.
\end{example}
\end{enumerate} \end{enumerate}
%TODO: Move this part. or move the third challenge some underneath. %TODO: Move this part. or move the third challenge some underneath.