Cleanup challenge 3
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Andreas Stadelmeier
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introduction.tex
120
introduction.tex
@ -552,12 +552,31 @@ is solved by removing the wildcard $\rwildcard{X}$ if possible.
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this problem is solved by ANF transformation
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\item \textbf{Wildcards as Existential Types:}
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One problem is the divergence between denotable and expressable types in Java \cite{semanticWildcardModel}.
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A wildcard in the Java syntax has no name and is bound to its enclosing type:
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$\exptype{List}{\exptype{List}{\type{?}}}$ equates to $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$.
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During type checking \emph{intermediate types}
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can emerge, which have no equivalent in the Java syntax.
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\item
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% This problem is solved by assuming everything is a wildcard and lateron erasing excessive wildcards
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% 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
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The program in listing \ref{shuffleExample} shows a challenge involving wildcards and subtyping.
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The method call \texttt{shuffle(l)} is incorrect, because \texttt{l} has the type
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$\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$ representing a list of unknown lists.
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Whereas $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$ is a subtype of
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$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$ representing a list of lists, all of the same type $\rwildcard{X}$,
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and can savely be passed to \texttt{shuffle}.
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This behavior can also be explained by looking at the types and their capture converted versions:
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\begin{center}
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\begin{tabular}{l | l | l}
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Java type & \TamedFJ{} representation & Capture Conversion \\
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\hline
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$\exptype{List}{\exptype{List}{\texttt{?}}}$ & $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$ & $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$ \\
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$\exptype{List2D}{\texttt{?}}$ & $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$ & $\exptype{List2D}{\rwildcard{X}}$ \\
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%Supertype of $\exptype{List2D}{\texttt{?}}$ & $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$ & $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$ \\
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\end{tabular}
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\end{center}
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%The direct supertype of $\exptype{List2D}{\rwildcard{X}}$ is $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$.
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%One problem is the divergence between denotable and expressable types in Java \cite{semanticWildcardModel}.
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%A wildcard in the Java syntax has no name and is bound to its enclosing type:
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%$\exptype{List}{\exptype{List}{\type{?}}}$ equates to $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$.
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%During type checking \emph{intermediate types}
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%can emerge, which have no equivalent in the Java syntax.
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\begin{lstlisting}[style=java,label=shuffleExample,caption=Intermediate Types Example]
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class List<X> extends Object {...}
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@ -571,45 +590,64 @@ List2D<?> l2d = ...;
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shuffle(l); // Error
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shuffle(l2d); // Valid
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\end{lstlisting}
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Java is using local type inference to allow method invocations which are not describable with regular Java types.
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The \texttt{shuffle} method in this case is invoked with the type $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$
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which is a subtype of $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$.
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After capture conversion \texttt{l2d'} has the type $\exptype{List}{\exptype{List}{\rwildcard{X}}}$
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and \texttt{shuffle} can be invoked with the type parameter $\rwildcard{X}$:
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\begin{lstlisting}[style=TamedFJ]
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let l2d' : (*@$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$@*) = l2d in <X>shuffle(l2d')
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\end{lstlisting}
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% Java is using local type inference to allow method invocations which are not describable with regular Java types.
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% The \texttt{shuffle} method in this case is invoked with the type $\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}}$
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% which is a subtype of $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$.
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% After capture conversion \texttt{l2d'} has the type $\exptype{List}{\exptype{List}{\rwildcard{X}}}$
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% and \texttt{shuffle} can be invoked with the type parameter $\rwildcard{X}$:
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% \begin{lstlisting}[style=TamedFJ]
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% let l2d' : (*@$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$@*) = l2d in <X>shuffle(l2d')
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% \end{lstlisting}
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For the example shown in listing \ref{shuffleExample} the method call \texttt{shuffle(l2d)} creates the constraints:
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\begin{constraintset}
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\begin{center}
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$
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\begin{array}{l}
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\wctype{\rwildcard{X}}{List2D}{\rwildcard{X}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}
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\\
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\hline
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\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}
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\\
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\hline
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\textit{Capture Conversion:}\
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\exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\wtv{x}}}
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\\
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\hline
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\textit{Solution:} \wtv{x} \doteq \rwildcard{X} \implies \exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\rwildcard{X}}}
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\end{array}
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$
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\end{center}
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\end{constraintset}
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% For the example shown in listing \ref{shuffleExample} the method call \texttt{shuffle(l2d)} creates the constraints:
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% \begin{constraintset}
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% \begin{center}
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% $
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% \begin{array}{l}
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% \wctype{\rwildcard{X}}{List2D}{\rwildcard{X}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}
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% \\
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% \hline
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% \wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}
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% \\
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% \hline
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% \textit{Capture Conversion:}\
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% \exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\wtv{x}}}
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% \\
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% \hline
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% \textit{Solution:} \wtv{x} \doteq \rwildcard{X} \implies \exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\rwildcard{X}}}
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% \end{array}
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% $
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% \end{center}
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% \end{constraintset}
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The method call \texttt{shuffle(l)} is invalid however,
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because \texttt{l} has the type
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$\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$.
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There is no solution for the subtype constraint:
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$\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}$
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Given such a program our type inference algorithm has to allow the call \texttt{shuffle(l2d)} and
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decline the call to \texttt{shuffle(l)}.
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% The method call \texttt{shuffle(l)} is invalid however,
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% because \texttt{l} has the type
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% $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$.
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% There is no solution for the subtype constraint:
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% $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\wtv{x}}}$
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\item \label{challenge3} \textbf{Free Variables cannot leaver their scope}:
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Let's assume we have a variable \texttt{ls} with type $\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$
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%When calling the \texttt{id} function with an element of this list we have to apply capture conversion.
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and the following input program:
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\noindent
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\begin{minipage}{0.62\textwidth}
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\begin{lstlisting}
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let x : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) = ls.get(0) in id(x) : (*@$\ntv{z}$@*)
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\end{lstlisting}\end{minipage}
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\hfill
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\begin{minipage}{0.36\textwidth}
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\begin{lstlisting}[style=constraints]
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(*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{x}}$@*),
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(*@$\exptype{List}{\wtv{x}} \lessdot \ntv{z},$@*)
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\end{lstlisting}
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\end{minipage}
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% the variable z has to
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\begin{example}
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Take the Java program in listing \ref{lst:mapExample} for example.
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It maps every element of a list
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$\expr{l} : \exptype{List}{\wctype{\rwildcard{A}}{List}{\rwildcard{A}}}$
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@ -657,8 +695,6 @@ $\exptype{List}{\wctype{\rwildcard{A}}{List}{\rwildcard{A}}}$
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We solve this by distinguishing between wildcard placeholders and normal placeholders.
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$\ntv{z}$ is a normal placeholder and is not allowed to contain free variables.
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\end{example}
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\end{enumerate}
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%TODO: Move this part. or move the third challenge some underneath.
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