572 lines
25 KiB
TeX
572 lines
25 KiB
TeX
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%TODO:
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% -Explain <c
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% - Kapitel 1.2 + 1.3 ist noch zu komplex
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% - Constraints und Unifikation erklären, bevor Unifikation erwähnt wird
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\section{Type Inference for Java}
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%The goal is to find a correct typing for a given Java program.
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Type inference for Java has many use cases and could be used to help programmers by inserting correct types for them,
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finding better type solutions for already typed Java programs (for example more generical ones),
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or allowing to write typeless Java code which is then type infered and thereby type checked by our algorithm.
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The algorithm proposed in this paper can determine a correct typing for the untyped Java source code example shown in figure \ref{fig:intro-example-typeless}.
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Our algorithm is also capable of finding solutions involving wildcards as shown in figure \ref{fig:intro-example-typed}.
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%This paper extends a type inference algorithm for Featherweight Java \cite{TIforFGJ} by adding wildcards.
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%The last step to create a type inference algorithm compatible to the Java type system.
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The algorithm presented in this paper is a slightly improved version of the one in \cite{TIforFGJ} including wildcard support.
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%a modified version of the \unify{} algorithm presented in \cite{plue09_1}.
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The input to the type inference algorithm is a Featherweight Java program (example in figure \ref{fig:nested-list-example-typeless}) conforming to the syntax shown in figure \ref{fig:syntax}.
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The \fjtype{} algorithm calculates constraints based on this intermediate representation,
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which are then solved by the \unify{} algorithm
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resulting in a correctly typed program (see figure \ref{fig:nested-list-example-typed}).
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\begin{itemize}
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\item
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We introduce the language \tifj{} (chapter \ref{sec:tifj}).
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A Featherweight Java derivative including Generics, Wildcards and Type Inference.
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\item
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We support capture conversion and Java style method calls.
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This requires existential types in a form which is not denotable by Java syntax \cite{aModelForJavaWithWildcards}.
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\item
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We present a novel approach to deal with existential types and capture conversion during constraint unification.
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The algorithm is split in two parts. A constraint generation step and an unification step.
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\item
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We proof soundness and aim for a good compromise between completeness and time complexity.
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\end{itemize}
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% Our algorithm finds a correct type solution for the following example, where the Java local type inference fails:
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% \begin{verbatim}
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% class SuperPair<A,B>{
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% A a;
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% B b;
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% }
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% class Pair<A,B> extends SuperPair<B,A>{
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% A a;
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% B b;
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% <X> X choose(X a, X b){ return b; }
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% String m(List<? extends Pair<Integer, String>> a, List<? extends Pair<Integer, String>> b){
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% return choose(choose(a,b).value.a,b.value.b);
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% }
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% }
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% \end{verbatim}
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\begin{figure}%[tp]
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\begin{subfigure}[t]{\linewidth}
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\begin{lstlisting}[style=fgj]
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class List<A> {
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List<A> add(A v) { ... }
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}
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class Example {
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m(l, la, lb){
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return l
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.add(la.add(1))
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.add(lb.add("str"));
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}
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}
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\end{lstlisting}
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\caption{Java method with missing return type}
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\label{fig:nested-list-example-typeless}
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\end{subfigure}
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~
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% \begin{subfigure}[t]{\linewidth}
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% \begin{lstlisting}[style=tfgj]
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% class List<A> {
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% List<A> add(A v) { ... }
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% }
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% class Example {
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% m(l, la, lb){
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% return let r2 : (*@$\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}$@*) = {
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% let r1 : (*@$\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}$@*) = l in {
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% let p1 : (*@$\exptype{List}{\type{Integer}}$@*) = {
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% let xa = la in xa.add(1)
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% } in x1.add(p1)
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% } in {
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% let p2 = {
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% let xb = lb in xb.add("str")
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% } in x2.add(p2)
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% };
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% }
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% }
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% \end{lstlisting}
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% \caption{Featherweight Java Representation}
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% \label{fig:nested-list-example-let}
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% \end{subfigure}
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% ~
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\begin{subfigure}[t]{\linewidth}
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\begin{lstlisting}[style=tfgj]
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class List<A> {
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List<A> add(A v) { ... }
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}
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class Example {
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m(List<List<? extends Object>> l, List<Integer> la, List<String> lb){
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return l
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.add(la.add(1))
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.add(lb.add("str"));
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}
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}
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\end{lstlisting}
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\caption{Java Representation}
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\label{fig:nested-list-example-typed}
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\end{subfigure}
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%\caption{Example code}
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%\label{fig:intro-example-code}
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\end{figure}
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%TODO
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% The goal is to proof soundness in respect to the type rules introduced by \cite{aModelForJavaWithWildcards}
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% and \cite{WildcardsNeedWitnessProtection}.
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\begin{figure}%[tp]
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\begin{subfigure}[t]{0.49\linewidth}
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\begin{lstlisting}[style=fgj]
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genList() {
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if( ... ) {
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return new List<String>();
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} else {
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return new List<Integer>();
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}
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}
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\end{lstlisting}
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\caption{Java method with missing return type}
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\label{fig:intro-example-typeless}
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\end{subfigure}
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~
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\begin{subfigure}[t]{0.49\linewidth}
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\begin{lstlisting}[style=tfgj]
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List<?> genList() {
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if( ... ) {
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return new List<String>();
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} else {
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return new List<Integer>();
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}
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}
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\end{lstlisting}
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\caption{Correct type}
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\label{fig:intro-example-typed}
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\end{subfigure}
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%\caption{Example code}
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%\label{fig:intro-example-code}
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\end{figure}
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% \subsection{Wildcards}
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% Java subtyping involving generics is invariant.
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% For example \texttt{List<String>} is not a subtype of \texttt{List<Object>}.
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% %Wildcards introduce variance by allowing \texttt{List<String>} to be a subtype of \texttt{List<?>}.
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% \texttt{List<Object>} is not a valid return type for the method \texttt{genList}.
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% The type inference algorithm has to find the correct type involving wildcards (\texttt{List<?>}).
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\subsection{Java Wildcards}
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Wildcards are expressed by a \texttt{?} in Java and can be used as type parameters.
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Wildcards add variance to Java type parameters.
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Generally Java has invariant subtyping for polymorphic types.
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A \texttt{List<String>} is not a subtype of \texttt{List<Object>} for example
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even though it seems intuitive with \texttt{String} being a subtype of \texttt{Object}.
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Wildcards can be formalized as existential types \cite{WildFJ}.
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\texttt{List<? extends Object>} and \texttt{List<? super String>} are both wildcard types
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denoted in our syntax by $\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}$ and
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$\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}}$.
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The syntax used here allows for wildcard parameters to have a name, an uppper and lower bound at the same time,
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and a type they are bound to.
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In this case the name is $\rwildcard{X}$ and it's bound to the the type \texttt{List}.
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Those properties are needed to formalize capture conversion.
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Polymorphic method calls need to be wraped in a process which \textit{opens} existential types \cite{addingWildcardsToJava}.
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In Java this is done implicitly in a process called capture conversion.
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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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%like $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}}$
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%or $\wctype{\rwildcard{X}}{Pair}{\rwildcard{X}, \rwildcard{X}}$
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can emerge, which have no equivalent in the Java syntax.
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%Our type inference algorithm uses existential types internally but spawns type solutions compatible with Java.
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Example: % This program is not typable with the Type Inference algorithm from Plümicke
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\begin{verbatim}
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class List<X> extends Object {...}
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class List2D<X> extends List<List<X>> {...}
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<X> void shuffle(List<List<X>> list) {...}
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List<List<?>> l = ...;
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List2D<?> l2d = ...;
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shuffle(l); // Error
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shuffle(l2d); // Valid
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\end{verbatim}
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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=letfj]
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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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The respective constraints are:
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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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\texttt{l} however has the type
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$\exptype{List}{\wctype{\rwildcard{X}}{List}{\rwildcard{X}}}$.
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The method call \texttt{shuffle(l)} is not correct, because 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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% $\exptype{List}{String} <: \wctype{\wildcard{X}{\bot}{\type{Object}}}{List}{\rwildcard{X}}$
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% means \texttt{List<String>} is a subtype of \texttt{List<? extend Object>}.
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\subsection{Global Type Inference}
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% A global type inference algorithm works on an input with no type annotations at all.
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% \begin{verbatim}
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% m(l) {
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% return l.add(l);
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% }
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% \end{verbatim}
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% $\tv{l} \lessdotCC \exptype{List}{\wtv{x}}, \tv{l} \lessdotCC \wtv{x}$
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\begin{description}
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\item[input] \tifj{} program
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\item[output] type solution
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\item[postcondition] the type solution applied to the input must yield a valid \letfj{} program
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\end{description}
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The input to our type inference algorithm is a modified version of the \letfj{}\cite{WildcardsNeedWitnessProtection} calculus (see chapter \ref{sec:tifj}).
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First \fjtype{} generates constraints
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and afterwards \unify{} computes a solution for the given constraint set.
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Constraints consist out of subtype constraints $(\type{T} \lessdot \type{T})$ and capture constraints $(\type{T} \lessdotCC \type{T})$.
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\textit{Note:} a type $\type{T}$ can also be a type placeholders $\ntv{a}$ or a wildcard type placeholder $\wtv{a}$.
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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}.
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\textit{Example:} $\exptype{List}{\ntv{a}} \lessdot \exptype{List}{\type{String}}$ is fulfilled by replacing type placeholder $\ntv{a}$ with the type $\type{String}$.
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Subtype constraints and type placeholders act the same as the ones used in \emph{Type Inference for Featherweight Generic Java} \cite{TIforFGJ}.
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The novel capture constraints and wildcard placeholders are needed for method invocations involving wildcards.
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%show input and a correct letFJ representation
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Even in a full typed program local type inference can be necessary.
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%TODO: first show local type inference and explain lessdotCC constraints. then show example with global TI
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Let's start with an example where all types are already given:
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\begin{lstlisting}[style=tfgj, caption=Valid Java program, label=lst:addExample]
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<A> List<A> add(List<A> l, A v)
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List<? super String> l = ...;
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add(l, "String");
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\end{lstlisting}
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\begin{lstlisting}[style=letfj, caption=\letfj{} representation of \texttt{add(l, "String")}, label=lst:addExampleLet]
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let l2 : (*@$\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}}$@*) = l in <X>add(l2, "String");
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\end{lstlisting}
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In \letfj{} there is no local type inference and all type parameters for a method call are mandatory.
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If wildcards are involved the so called capture conversion has to be done manually via let statements.
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A let statement \emph{opens} an existential type.
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In the body of the let statement the \textit{capture type} $\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}}$
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becomes $\exptype{List}{\rwildcard{X}}$ and the wildcard $\wildcard{X}{\type{Object}}{\type{String}}$ is free and can be used as
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a type parameter to \texttt{<X>add(...)}.
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%This is a valid Java program where the type parameters for the polymorphic method \texttt{add}
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%are determined by local type inference.
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Our type inference algorithm has to add let statements if necessary, including the capture types.
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Lets have a look at the constraints generated by \fjtype{} for the example in listing \ref{lst:addExample}:
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\begin{constraintset}
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\begin{center}
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$\begin{array}{c}
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\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}, \, \type{String} \lessdotCC \wtv{a}
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\\
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\hline
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\textit{Capture Conversion:}\ \exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}, \, \type{String} \lessdot \wtv{a}
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\\
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\hline
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\textit{Solution:}\ \wtv{a} \doteq \rwildcard{X} \implies \exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\rwildcard{X}}, \, \type{String} \lessdot \rwildcard{X}
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\end{array}
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$
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\end{center}
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\end{constraintset}
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%Why do we need the lessdotCC constraints here?
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The type of \texttt{l} can be capture converted by a let statement if needed (see listing \ref{lst:addExampleLet}).
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Therefore we assign the constraint $\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}$
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which allows \unify{} to do a capture conversion to $\exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$.
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\textit{Note:} The constraint $\type{String} \lessdot \rwildcard{X}$ is satisfied because $\rwildcard{X}$ has $\type{String}$ as lower bound.
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% No capture conversion for methods in the same class:
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% Given two methods without type annotations like
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% \begin{verbatim}
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% // m :: () -> r
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% m() = new List<String>() :? new List<Integer>();
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% // id :: (a) -> a
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% id(a) = a
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% \end{verbatim}
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% and a method call \texttt{id(m())} would lead to the constraints:
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% $\exptype{List}{\type{String}} \lessdot \ntv{r},
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% \exptype{List}{\type{Integer}} \lessdot \ntv{r},
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% \ntv{r} \lessdotCC \ntv{a}$
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% In this example capture conversion is not applicable,
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% because the \texttt{id} method is not polymorphic.
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% The type solution provided by \unify{} for this constraint set is:
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% $\sigma(\ntv{r}) = \wctype{\rwildcard{X}}{List}{\rwildcard{X}},
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% \sigma(\ntv{a}) = \wctype{\rwildcard{X}}{List}{\rwildcard{X}}$
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% \begin{verbatim}
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% List<?> m() = new List<String>() :? new List<Integer>();
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% List<?> id(List<?> a) = a
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% \end{verbatim}
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The following example has the \texttt{id} method already typed and the method \texttt{m}
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extended by a recursive call \texttt{id(m())}:
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\begin{verbatim}
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<A> List<A> id(List<A> a) = a
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m() = new List<String>() :? new List<Integer>() :? id(m());
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\end{verbatim}
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Now the constraints make use of a $\lessdotCC$ constraint:
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$\exptype{List}{\type{String}} \lessdot \ntv{r},
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\exptype{List}{\type{Integer}} \lessdot \ntv{r},
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\ntv{r} \lessdotCC \exptype{List}{\wtv{a}}$
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After substituting $\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ for $\ntv{r}$ like in the example before
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we get the constraint $\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}$.
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Due to the $\lessdotCC$ \unify{} is allowed to perform a capture conversion yielding
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$\exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$.
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\textit{Note:} The wildcard placeholder $\wtv{a}$ is allowed to hold free variables whereas a normal placeholder like $\ntv{r}$
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is never assigned a type containing free variables.
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Therefore \unify{} sets $\wtv{a} \doteq \rwildcard{X}$, completing the constraint set and resulting in the type solution:
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\begin{verbatim}
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List<?> m() = new List<String>() :? new List<Integer>() :? id(m());
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\end{verbatim}
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\subsection{Challenges}\label{challenges}
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%TODO: Wildcard subtyping is infinite see \cite{TamingWildcards}
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The introduction of wildcards adds additional challenges.
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% we cannot replace every type variable with a wildcard
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Type variables can also be used as type parameters, for example
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$\exptype{List}{String} \lessdot \exptype{List}{\tv{a}}$.
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A problem arises when replacing type variables with wildcards.
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% Wildcards are not reflexive.
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% ( on the equals property ), every wildcard has to be capture converted when leaving its scope
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% do not substitute free type variables
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Lets have a look at two examples:
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\begin{itemize}
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\item \begin{example} \label{intro-example1}
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The first one is a valid Java program.
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The type \texttt{List<? super String>} is \textit{capture converted} to a fresh type variable $\rwildcard{X}$
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which is used as the generic method parameter \texttt{A}.
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Java uses capture conversion to replace the generic \texttt{A} by a capture converted version of the
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\texttt{? super String} wildcard.
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Knowing that the type \texttt{String} is a subtype of any type the wildcard \texttt{? super String} can inherit
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it is safe to pass \texttt{"String"} for the first parameter of the function.
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\begin{verbatim}
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<A> List<A> add(List<A> l, A v) {}
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List<? super String> list = ...;
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add("String", list);
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\end{verbatim}
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\end{example}
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\item \begin{example}\label{intro-example2}
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This example displays an incorrect Java program.
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The method call to \texttt{concat} with two wildcard lists is unsound.
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Each list could be of a different kind and therefore the \texttt{concat} cannot succeed.
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\begin{verbatim}
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<A> List<A> concat(List<A> l1, List<A> l2) { ... }
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List<?> list = ... ;
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concat(list, list);
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\end{verbatim}
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% $\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}, \\
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% \wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$
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\end{example}
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% \item
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% \unify{} morphs a constraint set into a correct type solution
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% gradually assigning types to type placeholders during that process.
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% Solved constraints are removed and never considered again.
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% In the following example \unify{} solves the constraint generated by the expression
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% \texttt{l.add(l.head())} first, which results in $\ntv{l} \lessdot \exptype{List}{\wtv{a}}$.
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% \begin{verbatim}
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% anyList() = new List<String>() :? new List<Integer>()
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% add(anyList(), anyList().head());
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% \end{verbatim}
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% The type for \texttt{l} can be any kind of list, but it has to be a invariant one.
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% Assigning a \texttt{List<?>} for \texttt{l} is unsound, because the type list hiding behind
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% \texttt{List<?>} could be a different one for the \texttt{add} call than the \texttt{head} method call.
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% An additional constraint $\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$
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% is solved by removing the wildcard $\rwildcard{X}$ if possible.
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\end{itemize}
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The \unify{} algorithm only sees the constraints with no information about the program they originated from.
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The main challenge was to find an algorithm which computes $\sigma(\wtv{a}) = \rwildcard{X}$ for example \ref{intro-example1} but not for example \ref{intro-example2}.
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\subsection{Capture Conversion}
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The input to our type inference algorithm does not contain let statements.
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Those are added after computing a type solution.
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Let statements act as capture conversion and only have to be applied in method calls involving wildcard types.
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\begin{figure}
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\begin{minipage}{0.45\textwidth}
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\begin{lstlisting}[style=fgj]
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<X> List<X> clone(List<X> l);
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example(p){
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return clone(p);
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}
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\end{lstlisting}
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\end{minipage}%
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\hfill
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\begin{minipage}{0.5\textwidth}
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\begin{lstlisting}[style=tfgj]
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(*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) example((*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) p){
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return let x : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) = p in
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clone(x) : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*);
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}
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\end{lstlisting}
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\end{minipage}
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\caption{Type inference adding capture conversion}\label{fig:addingLetExample}
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\end{figure}
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Figure \ref{fig:addingLetExample} shows a let statement getting added to the typed output.
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The method \texttt{clone} cannot be called with the type $\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$.
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After a capture conversion \texttt{x} has the type $\exptype{List}{\rwildcard{X}}$ with $\rwildcard{X}$ being a free variable.
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Afterwards we have to find a supertype of $\exptype{List}{\rwildcard{X}}$, which does not contain free variables
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($\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ in this case).
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During the constraint generation step most types are not known yet and are represented by a type placeholder.
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During a methodcall like the one in the \texttt{example} method in figure \ref{fig:ccExample} the type of the parameter \texttt{p}
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is not known yet.
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The type \texttt{List<?>} would be one possibility as a parameter type for \texttt{p}.
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To make wildcards work for our type inference algorithm \unify{} has to apply capture conversions if necessary.
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The type placeholder $\tv{r}$ is the return type of the \texttt{example} method.
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One possible type solution is $\tv{p} \doteq \tv{r} \doteq \wctype{\rwildcard{X}}{List}{\rwildcard{X}}$,
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which leads to:
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\begin{verbatim}
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List<?> example(List<?> p){
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return clone(p);
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}
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\end{verbatim}
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But by substituting $\tv{p} \doteq \wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ in the constraint
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$\tv{p} \lessdotCC \exptype{List}{\wtv{x}}$ leads to
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$\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{x}}$.
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To make this typing possible we have to introduce a capture conversion via a let statement:
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$\texttt{return}\ (\texttt{let}\ \texttt{x} : \wctype{\rwildcard{X}}{List}{\rwildcard{X}} = \texttt{p}\ \texttt{in} \
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\texttt{clone}\generics{\rwildcard{X}}(x) : \wctype{\rwildcard{X}}{List}{\rwildcard{X}})$
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Inside the let statement the variable \texttt{x} has the type $\exptype{List}{\rwildcard{X}}$
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This spawns additional problems.
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\begin{figure}
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\begin{minipage}{0.45\textwidth}
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\begin{verbatim}
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<X> List<X> clone(List<X> l){...}
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example(p){
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return clone(p);
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}
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\end{verbatim}
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\end{minipage}%
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\hfill
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\begin{minipage}{0.35\textwidth}
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\begin{constraintset}
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\textbf{Constraints:}\\
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$
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\tv{p} \lessdotCC \exptype{List}{\wtv{x}}, \\
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\tv{p} \lessdot \tv{r}, \\
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\tv{p} \lessdot \type{Object},
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\tv{r} \lessdot \type{Object}
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$
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\end{constraintset}
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\end{minipage}
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\caption{Type inference example}\label{fig:ccExample}
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\end{figure}
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In addition with free variables this leads to unwanted behaviour.
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Take the constraint
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$\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$ for example.
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After a capture conversion from $\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$ to $\exptype{List}{\rwildcard{Y}}$ and a substitution $\wtv{a} \doteq \rwildcard{Y}$
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we get
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$\exptype{List}{\rwildcard{Y}} \lessdot \exptype{List}{\rwildcard{Y}}$.
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Which is correct if we apply capture conversion to the left side:
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$\exptype{List}{\rwildcard{X}} <: \exptype{List}{\rwildcard{X}}$
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If the input constraints did not intend for this constraint to undergo a capture conversion then \unify{} would produce an invalid
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type solution due to:
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$\wctype{\rwildcard{X}}{List}{\rwildcard{X}} \nless: \exptype{List}{\rwildcard{X}}$
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The reason for this is the \texttt{S-Exists} rule's premise
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$\text{dom}(\Delta') \cap \text{fv}(\exptype{List}{\rwildcard{X}}) = \emptyset$.
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Additionally free variables are not allowed to leave the scope of a capture conversion
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introduced by a let statement.
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%TODO we combat both of this with wildcard type placeholders (? flag)
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Type placeholders which are not flagged as possible free variables ($\wtv{a}$) can never hold a free variable or a type containing free variables.
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Constraint generation places these standard place holders at method return types and parameter types.
|
|
\begin{lstlisting}[style=fgj]
|
|
<X> List<X> clone(List<X> l);
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(*@$\red{\tv{r}}$@*) example((*@$\red{\tv{p}}$@*) p){
|
|
return clone(p);
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|
}
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\end{lstlisting}
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This prevents type solutions that contain free variables in parameter and return types.
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When calling a method which already has a type annotation we have to consider adding a capture conversion in form of a let statement.
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The constraint $\tv{p} \lessdot \exptype{List}{\wtv{x}}$ signals the \unify{} algorithm that here a capture conversion is possible.
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|
$\sigma(\tv{p}) = \wctype{\rwildcard{X}}{List}{\rwildcard{X}}, \sigma(\tv{r}) = \wctype{\rwildcard{X}}{List}{\rwildcard{X}}, $ is a possible solution.
|
|
But only when adding a capture conversion:
|
|
\begin{lstlisting}[style=fgj]
|
|
<X> List<X> clone(List<X> l);
|
|
(*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) example((*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) p){
|
|
return let x : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*) = p in clone(x) : (*@$\wctype{\rwildcard{X}}{List}{\rwildcard{X}}$@*);
|
|
}
|
|
\end{lstlisting}
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|
|
|
Capture constraints cannot be stored in a set.
|
|
$\wtv{a} \lessdotCC \wtv{b}$ is not the same as $\wtv{a} \lessdotCC \wtv{b}$.
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|
Both constraints will end up the same after a substitution for both placeholders $\tv{a}$ and $\tv{b}$.
|
|
But afterwards a capture conversion is applied, which can generate different types on the left sides.
|
|
\begin{itemize}
|
|
\item $\text{CC}(\wctype{\rwildcard{X}}{List}{\rwildcard{X}}) \implies \exptype{List}{\rwildcard{Y}}$
|
|
\item $\text{CC}(\wctype{\rwildcard{X}}{List}{\rwildcard{X}}) \implies \exptype{List}{\rwildcard{Z}}$
|
|
\end{itemize} |