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JanUlrich 2024-04-29 15:38:01 +02:00
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@ -62,7 +62,7 @@
%frame=single, % Rahmen an %frame=single, % Rahmen an
%frameround=ffff, %frameround=ffff,
rulecolor=\color{darkgray}, % Rahmenfarbe rulecolor=\color{darkgray}, % Rahmenfarbe
fillcolor=\color{ListingBackground} fillcolor=\color{ListingBackground},
showtabs=false, showtabs=false,
%breaklines=true, %breaklines=true,
%breakatwhitespace=true, %breakatwhitespace=true,
@ -213,243 +213,45 @@
\newcommand{\rulenameAfter}[1]{\begin{array}[b]{l}\rulename{#1}\\[-0.5em] \ \end{array}} \newcommand{\rulenameAfter}[1]{\begin{array}[b]{l}\rulename{#1}\\[-0.5em] \ \end{array}}
\begin{frame}[fragile] \begin{frame}[fragile]{Global Type Inference}
\frametitle{Java Method Call} \begin{lstlisting}
\rulename{T-Call} \\[1em] (*@\only<1>{\color{gray}\texttt{List<? extends Object>} }@*)someList(){
\begin{center} if(Math.random > 0.5){
$\begin{array}{l} return new List(*@\only<1>{\color{gray}\texttt{<String>}}@*)("String");
\begin{array}{@{}c} } else {
\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad return new List(*@\only<1>{\color{gray}\texttt{<Integer>}}@*)(42);
\ol{v} : [\ol{T'}/\ol{X}]\ol{T}
\\
\hline
\vspace*{-0.3cm}\\
\generics{\ol{T'}}\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
\end{array}
\end{array}$
\end{center}
\textbf{Example:}
$\generics{\type{String}}\texttt{emptyList}() : \exptype{List}{\type{String}}$
\\[2em]
\rulename{TI-Call} \\[1em]
\begin{center}
$\begin{array}{l}
\begin{array}{@{}c}
\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
\ol{v} : [\ol{T'}/\ol{X}]\ol{T}
\\
\hline
\vspace*{-0.3cm}\\
\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
\end{array}
\end{array}$
\end{center}
\textbf{Example:}
$\texttt{emptyList}() : \exptype{List}{\type{String}}$
\end{frame}
% \begin{frame}[fragile]
% \frametitle{Java Method Call}
% \rulename{M-Class} \\[1em]
% \begin{center}
% $\begin{array}{l}
% \begin{array}{@{}c}
% \generics{\ol{X}}\ \type{T} \ \texttt{m}(\ol{T\ x}) \set{ \ldots }
% \\
% \hline
% \vspace*{-0.3cm}\\
% \textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T}
% \end{array}
% \end{array}$
% \end{center}
% \textbf{Beispiele:}\\
% \begin{verbatim}
% X head(List<X> l) { ... }
% \end{verbatim}
% \begin{itemize}
% \item
% $\textit{mtype}(\texttt{head}) = \generics{\type{X}}\ \exptype{List}{\type{X}} \to \type{X}$
% \end{itemize}
% \begin{verbatim}
% <X> List<X> emptyList() { ... }
% \end{verbatim}
% \begin{itemize}
% \item
% $\textit{mtype}(\texttt{emptyList}) = \generics{\type{X}}\ [] \to \exptype{List}{\type{X}}$
% \end{itemize}
% \end{frame}
\begin{frame}[fragile]
\frametitle{Java Method Call}
\rulename{T-Call} \\[1em]
\begin{center}
$\begin{array}{l}
\begin{array}{@{}c}
\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
\Delta \vdash \expr{v}, \ol{v} : [\ol{T'}/\ol{X}]\ol{T}
\\
\hline
\vspace*{-0.3cm}\\
\Delta \vdash \expr{v}.\generics{\ol{T'}}\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
\end{array}
\end{array}$
\end{center}
\textbf{Beispiel:}\\
\begin{verbatim}
<X> X head(List<X> l){ ... }
head(new List<String>());
\end{verbatim}
\pause
$\begin{array}{l}
\begin{array}{@{}c}
\textit{mtype}(\texttt{head}) = \generics{\ol{X}}\ \exptype{List}{\type{X}} \to \type{X} \\
\texttt{new List<String>}() : [\type{String}/\type{X}]\exptype{List}{\type{X}}
\\
\hline
\vspace*{-0.3cm}\\
\generics{\type{String}}\texttt{head}(\texttt{new List<String>}()) : [\type{String}/\type{X}]\type{X}
\end{array}
\end{array}$\\[1em]
\pause
% $\begin{array}{l}
% \begin{array}{@{}c}
% \textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
% \texttt{new List<String>}() : \exptype{List}{\type{String}}
% \\
% \hline
% \vspace*{-0.3cm}\\
% \generics{\type{String}}\texttt{head}(\texttt{new List<String>}()) : \type{String}
% \end{array}
% \end{array}$
\begin{itemize}
\item \texttt{head(new List<String>()) : String}
\end{itemize}
\end{frame}
% \begin{frame}[fragile]
% \frametitle{Java Method Call}
% \begin{verbatim}
% head(head(emptyList()))
% \end{verbatim}
% $
% \begin{array}[b]{c}
% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
% \\
% \hline
% \texttt{head(emptyList())} : \exptype{List}{\type{String}}
% \end{array} \rulenameAfter{T-Call}
% $
% \end{frame}
% \begin{frame}[fragile]
% \frametitle{Java Method Call}
% $
% \begin{array}[b]{c}
% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
% \\
% \hline
% \generics{\exptype{List}{\exptype{List}{\type{String}}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
% \end{array} \rulenameAfter{T-Call}
% $
% $
% \begin{array}[b]{c}
% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
% \\
% \hline
% \generics{\exptype{List}{\type{String}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
% \end{array} \rulenameAfter{T-Call}
% $
% \end{frame}
% \begin{frame}[fragile]
% \frametitle{Java Method Call}
% $
% \begin{array}[b]{c}
% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
% \\
% \hline
% \generics{\exptype{List}{\type{String}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
% \end{array} \rulenameAfter{T-Call}
% $
% \end{frame}
% \begin{frame}[fragile]
% \frametitle{Java Method Call}
% $
% \begin{array}[b]{c}
% \begin{array}[b]{c}
% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}}
% \\
% \hline
% \texttt{emptyList().get()} : \exptype{List}{\type{String}}
% \end{array} \rulenameAfter{T-Call}
% \\
% \hline
% \texttt{emptyList().get().get()} : \type{String}
% \end{array} \rulenameAfter{T-Call}
% $
% \end{frame}
% \begin{frame}[fragile]
% \frametitle{Method Call with Wildcards}
% Eigentlich kann man mit Wildcard Typen keine Methodenaufrufe durchführen.
% Der eigentliche Typ ist nicht bekannt.
% ? sind auch nicht reflexiv
% Es geht dennoch, weil sich hinter einem Wildcard Typ ein konkreter Typ verbirgt.
% Dieser muss ausgepackt werden.
% Die entstehende freie Typvariable kann nun als Typ benutzt werden.
% \end{frame}
\begin{frame}[fragile]
\frametitle{Typeinference ohne Wildcards}
\rulename{TI-Call} \\[1em]
\begin{center}
$\begin{array}{l}
\begin{array}{@{}c}
\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
\Delta \vdash \expr{v}, \ol{v} : [\ol{T'}/\ol{X}]\ol{T}
\\
\hline
\vspace*{-0.3cm}\\
\Delta \vdash \expr{v}.\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
\end{array}
\end{array}$
\end{center}
\begin{verbatim}
untypedMethod(l){
shuffle(l);
} }
\end{verbatim} }
\end{lstlisting}
\pause
\end{frame}
\begin{frame}[fragile]{Wildcards as Existential Types}
\begin{lstlisting}
(*@\only<2>{\color{red}$\wctype{\wildcard{A}{\type{Object}}{\bot}}{List}{\rwildcard{A}}$ }@*)someList(){
if(Math.random > 0.5){
return new List("String");
} else {
return new List(42);
}
}
\end{lstlisting}
\pause
\begin{itemize} \begin{itemize}
\item $\lessdot $ \item \texttt{List<? extends Object>} $\implies$ $\wctype{\wildcard{A}{\type{Object}}{\bot}}{List}{\rwildcard{A}}$
\item \texttt{List<? super String>} $\implies$ $\wctype{\wildcard{A}{\type{Object}}{\type{String}}}{List}{\rwildcard{A}}$
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\begin{frame}[fragile]{Capture Conversion} \begin{frame}[fragile]{Capture Conversion}
\rulename{T-Call-Wildcards} \\[1em] \rulename{T-Call-Wildcards} \\[1em]
\alt<-2>{
\begin{center} \begin{center}
$\begin{array}{l} $\begin{array}{l}
\begin{array}{@{}c} \begin{array}{@{}c}
\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad \textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
\Delta \vdash \ol{v} : \ol{\wctype{\Delta}{C}{\ol{T}}} \\ \Delta \vdash \ol{v} : \ol{\wctype{\Delta}{C}{\ol{T}}} \\
\overline{\exptype{C}{\ol{T}}} <: [\ol{T'}/\ol{X}]\ol{T} \quad \quad [\ol{T'}/\ol{X}]\type{T} <: \type{R} \Delta, \overline{\Delta} \vdash \overline{\exptype{C}{\ol{T}}} <: [\ol{T'}/\ol{X}]\ol{T} \quad \quad \Delta, \overline{\Delta} \vdash [\ol{T'}/\ol{X}]\type{T} <: \type{R}
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
@ -457,45 +259,35 @@ $\begin{array}{l}
\end{array} \end{array}
\end{array}$ \end{array}$
\end{center} \end{center}
%Wildcard Capture }{
\begin{verbatim}
<X> List<X> shuffle(List<X> l){ ... }
List<?> l;
shuffle(l) : List<?>
\end{verbatim}
%why do we need capture constraints?
% List<List<?>> not a subtype of List<List<X>> for any X
\begin{center} \begin{center}
$\begin{array}{l} $\begin{array}{l}
\begin{array}{@{}c} \begin{array}{@{}c}
\textit{mtype}(\texttt{shuffle}) = \generics{\type{X}}\ \exptype{List}{\type{X}} \to \exptype{List}{\type{X}} \\ \textit{mtype}(\texttt{shuffle}) = \generics{\type{X}}\ \exptype{List}{\type{X}} \to \exptype{List}{\type{X}} \quad \quad
\texttt{l} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \quad \quad \texttt{l} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \\
\exptype{List}{\type{A}} <: [\type{A}/\type{X}]\exptype{List}{\type{X}} \\ \wildcard{A}{\type{Object}}{\bot} \vdash \exptype{List}{\type{A}} <: [\type{A}/\type{X}]\exptype{List}{\type{X}} \quad \quad
\ [\type{A}/\type{X}]\exptype{List}{\type{X}} <: \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \ [\type{A}/\type{X}]\exptype{List}{\type{X}} <: \wctype{\rwildcard{B}}{List}{\rwildcard{B}}
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\texttt{shuffle}(\expr{l}) : \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \texttt{shuffle}(\expr{l}) : \wctype{\rwildcard{B}}{List}{\rwildcard{N}}
\end{array} \end{array}
\end{array}$ \end{array}$
\end{center} \end{center}
\end{frame}
\begin{frame}[fragile,t]
%Explain why ? cannot be used as a regular type
\begin{verbatim}
<X> List<X> concat(List<X> l1, List<X> l2){
return l1.addAll(l2);
} }
\pause
%Wildcard Capture
\begin{lstlisting}
<X> List<X> shuffle(List<X> l){ ... }
List<?> l1; List<? extends Object> l;
List<?> l2;
shuffle(l) (*@\only<3>{\color{red}\texttt{: List<? extends Object>}}@*)
\end{lstlisting}
%why do we need capture constraints?
% List<List<?>> not a subtype of List<List<X>> for any X
\pause
concat(l1, l2); //Error
\end{verbatim}
\end{frame} \end{frame}
\begin{frame}[fragile,t] \begin{frame}[fragile,t]
@ -514,138 +306,101 @@ concat(l1, l2); //Error (würde l2 an ls anfügen!)
\end{verbatim} \end{verbatim}
\end{frame} \end{frame}
\begin{frame}[fragile] \begin{frame}[fragile,t]
%Wildcard Capture %Explain why ? cannot be used as a regular type
\begin{verbatim} \begin{verbatim}
<X> void shuffle2D(List<List<X>> l){ <X> List<X> concat(List<X> l1, List<X> l2){
//randomly exchange every element return l1.addAll(l2);
// in the two dimensional list l
} }
List<List<?>> l; List<?> l1;
List<?> l2;
shuffle2D(l); // Error concat(l1, l2); //Error
\end{verbatim} \end{verbatim}
$\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \nless: \exptype{List}{\rwildcard{A}}$
\end{frame}
\begin{frame}[fragile]
%Es gibt stellen wo capture conversion möglich ist und Stellen wo es nicht ist.
\frametitle{Capture Constraint}
\begin{verbatim}
<X> List<List<X>> shuffle2D(List<? extends List<X>> l) {...}
List<List<?>> l = ...;
shuffle2D(l); // Fehler!
\end{verbatim}
\begin{itemize}
\item hier ergibt sich ein Subtype-Constraint der keine Capture Conversion zulässt:
\item $\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{x}}$
\pause \pause
\item $\unify{}(\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{a}}) = \emptyset$ \vfill
\end{itemize} \begin{center}
$\begin{array}{l}
\end{frame} \begin{array}{@{}c}
\textit{mtype}(\texttt{concat}) = \generics{\type{X}}\ \exptype{List}{\type{X}} \to \exptype{List}{\type{X}} \to \exptype{List}{\type{X}} \\
\begin{frame}[fragile] \texttt{l1} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}}, \texttt{l2} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \\
%Es gibt stellen wo capture conversion möglich ist und Stellen wo es nicht ist. \exptype{List}{\rwildcard{A}} \nless: [{\color{red}?}/\type{X}]\exptype{List}{\type{X}} \quad \quad
\frametitle{Capture Constraint} \exptype{List}{\rwildcard{B}} \nless: [{\color{red}?}/\type{X}]\exptype{List}{\type{X}}
\begin{verbatim}
<X> List<List<X>> shuffle2D(List<? extends List<X>> l) {...}
List<List<?>> l = ...;
shuffle2D(l); // Fehler!
\end{verbatim}
\begin{itemize}
\item hier ergibt sich ein Subtype-Constraint der keine Capture Conversion zulässt:
\item $\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{x}}$
\pause
\item $\unify{}(\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{a}}) = \emptyset$
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Capture Constraint}
$\unify{}(\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{a}}) = \emptyset$
$\unify{}(\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{a}}) = \set{\tv{a} \to \rwildcard{A}}$
\end{frame}
% TODO:
% Folie zu aktueller Stand Unify: Es gibt <. Constraints. Unify löst diese
% Folie zu <? constraints: sie werden benötigt, weil unify während der Lösungsfindung Capture Conversion durchführen muss
%why do we need capture constraints?
% List<List<?>> not a subtype of List<List<X>> for any X
%TODO Unify erklären
% - es gibt die Capture Regel, welche Capture Conversion durchführt
% - die subst-Regel, welche keine Typen mit freien Variablen in normale Typvariablen einsetzt
\begin{frame}[fragile]{Reduce}
\rulename{Reduce} $
\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \vdash
C \cup \, \set{ \exptype{C}{\ol{S}} \lessdot
\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}} } \\
\hline
\vspace*{-0.4cm}\\
\wildcardEnv
\vdash C \cup \, \set{
\ol{\type{S}} \doteq [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{\type{T}},
\overline{\wtv{a} \lessdot \type{U}}, \overline{\type{L} \lessdot \wtv{a}}}
\end{array}
%\quad \ol{Y} = \textit{fresh}(\ol{X})
\quad \begin{array}[c]{l}
\ol{\wtv{a}} \ \text{fresh}\\
%\text{fv}(\exptype{C}{\ol{S}}) \subseteq \text{dom}(\overline{\wildcard{B}{\type{U'}}{\type{L'}}})
%\text{dom}(\overline{\wildcard{A}{\type{U}}{\type{L}}}) \subseteq \text{fv}(\exptype{C}{\ol{T}}) \\
%\text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}) = \emptyset
\end{array}
\end{array}
$\\[1em]
\begin{itemize}
\item \textbf{Example:}
$\begin{array}[t]{c}
\exptype{List}{\type{String}} \lessdot \wctype{\wildcard{A}{\type{Object}}{\bot}}{List}{\type{A}}\\
\hline
\type{String} \doteq \wtv{a}, \wtv{a} \lessdot \type{Object}, \bot \lessdot \wtv{a}
\end{array}$
\end{itemize}\\ \vfill
\pause
\rulename{Capture} $
\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \vdash
C \cup \, \set{ \wctype{\overline{\wildcard{B}{\type{U}}{\type{L}}}}{C}{\ol{S}} \lessdotCC \type{T} } \\
\hline
\vspace*{-0.4cm}\\
\wildcardEnv \cup \overline{\wildcard{C}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{U}}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{L}}}
\vdash C \cup \, \set{
[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}] \exptype{C}{\ol{S}} \lessdot \type{T} }
\end{array}
%\quad \ol{Y} = \textit{fresh}(\ol{X})
\quad \begin{array}[c]{l}
\ol{\rwildcard{C}} \ \text{fresh}\\
%\text{fv}(\type{T}) \neq \emptyset
\end{array}
\end{array}
$\\[1em]
\begin{itemize}
\item \textbf{Example:}
$\begin{array}[t]{c}
\wctype{\wildcard{A}{\type{U}}{\type{L}}}{List}{\type{A}} \lessdot \exptype{List}{\wtv{x}}\\
\hline
\wildcard{A}{\type{U}}{\type{L}} \vdash \exptype{List}{\type{A}} \lessdot \exptype{List}{\wtv{x}}
\\ \\
\hline \hline
\wildcard{A}{\type{U}}{\type{L}} \vdash \type{A} \doteq \wtv{x} \vspace*{-0.3cm}\\
\texttt{shuffle}(\expr{l}) : {\color{red}\textbf{Errror}}
\end{array}
\end{array}$ \end{array}$
\end{itemize} \end{center}
\end{frame} \end{frame}
% - es gibt die Capture Regel, welche Capture Conversion durchführt
% - die subst-Regel, welche keine Typen mit freien Variablen in normale Typvariablen einsetzt
% \begin{frame}[fragile]{Reduce}
% \rulename{Reduce} $
% \begin{array}[c]{@{}ll}
% \begin{array}[c]{l}
% \wildcardEnv \vdash
% C \cup \, \set{ \exptype{C}{\ol{S}} \lessdot
% \wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}} } \\
% \hline
% \vspace*{-0.4cm}\\
% \wildcardEnv
% \vdash C \cup \, \set{
% \ol{\type{S}} \doteq [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{\type{T}},
% \overline{\wtv{a} \lessdot \type{U}}, \overline{\type{L} \lessdot \wtv{a}}}
% \end{array}
% %\quad \ol{Y} = \textit{fresh}(\ol{X})
% \quad \begin{array}[c]{l}
% \ol{\wtv{a}} \ \text{fresh}\\
% %\text{fv}(\exptype{C}{\ol{S}}) \subseteq \text{dom}(\overline{\wildcard{B}{\type{U'}}{\type{L'}}})
% %\text{dom}(\overline{\wildcard{A}{\type{U}}{\type{L}}}) \subseteq \text{fv}(\exptype{C}{\ol{T}}) \\
% %\text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}) = \emptyset
% \end{array}
% \end{array}
% $\\[1em]
% \begin{itemize}
% \item \textbf{Example:}
% $\begin{array}[t]{c}
% \exptype{List}{\type{String}} \lessdot \wctype{\wildcard{A}{\type{Object}}{\bot}}{List}{\type{A}}\\
% \hline
% \type{String} \doteq \wtv{a}, \wtv{a} \lessdot \type{Object}, \bot \lessdot \wtv{a}
% \end{array}$
% \end{itemize}\\ \vfill
% \pause
% \rulename{Capture} $
% \begin{array}[c]{@{}ll}
% \begin{array}[c]{l}
% \wildcardEnv \vdash
% C \cup \, \set{ \wctype{\overline{\wildcard{B}{\type{U}}{\type{L}}}}{C}{\ol{S}} \lessdotCC \type{T} } \\
% \hline
% \vspace*{-0.4cm}\\
% \wildcardEnv \cup \overline{\wildcard{C}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{U}}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{L}}}
% \vdash C \cup \, \set{
% [\ol{\rwildcard{C}}/\ol{\rwildcard{B}}] \exptype{C}{\ol{S}} \lessdot \type{T} }
% \end{array}
% %\quad \ol{Y} = \textit{fresh}(\ol{X})
% \quad \begin{array}[c]{l}
% \ol{\rwildcard{C}} \ \text{fresh}\\
% %\text{fv}(\type{T}) \neq \emptyset
% \end{array}
% \end{array}
% $\\[1em]
% \begin{itemize}
% \item \textbf{Example:}
% $\begin{array}[t]{c}
% \wctype{\wildcard{A}{\type{U}}{\type{L}}}{List}{\type{A}} \lessdot \exptype{List}{\wtv{x}}\\
% \hline
% \wildcard{A}{\type{U}}{\type{L}} \vdash \exptype{List}{\type{A}} \lessdot \exptype{List}{\wtv{x}}
% \\
% \hline
% \wildcard{A}{\type{U}}{\type{L}} \vdash \type{A} \doteq \wtv{x}
% \end{array}$
% \end{itemize}
% \end{frame}
\begin{frame}[fragile]{Wildcard Creation} \begin{frame}[fragile]{Wildcard Creation}
\begin{lstlisting} \begin{lstlisting}
(*@\only<2>{$\color{red}\tv{r}$ }@*)someList(){ (*@\only<2>{$\color{red}\tv{r}$ }@*)someList(){
@ -735,17 +490,80 @@ $
$ $
\end{frame} \end{frame}
\begin{frame}[fragile] \begin{frame}[fragile]{Reduce}
$ $
\begin{array}{l} \begin{array}{l}
\exptype{List}{String} \lessdot \highlight{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}}, \exptype{List}{String} \lessdot \highlight{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}},
\exptype{List}{Integer} \lessdot \highlight{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}} \exptype{List}{Integer} \lessdot \highlight{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}}
\\
\hline
\type{String} \doteq \wtv{x1}, \type{Integer} \doteq \wtv{x2}
\end{array} \end{array}
\rulenameAfter{Reduce} \rulenameAfter{Reduce}
$\\[1em]
\rulename{Reduce} $
\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \vdash
C \cup \, \set{ \exptype{C}{\ol{S}} \lessdot
\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}} } \\
\hline
\vspace*{-0.4cm}\\
\wildcardEnv
\vdash C \cup \, \set{
\ol{\type{S}} \doteq [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{\type{T}},
\overline{\wtv{a} \lessdot \type{U}}, \overline{\type{L} \lessdot \wtv{a}}}
\end{array}
%\quad \ol{Y} = \textit{fresh}(\ol{X})
\quad \begin{array}[c]{l}
\ol{\wtv{a}} \ \text{fresh}\\
%\text{fv}(\exptype{C}{\ol{S}}) \subseteq \text{dom}(\overline{\wildcard{B}{\type{U'}}{\type{L'}}})
%\text{dom}(\overline{\wildcard{A}{\type{U}}{\type{L}}}) \subseteq \text{fv}(\exptype{C}{\ol{T}}) \\
%\text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}) = \emptyset
\end{array}
\end{array}
$\\[1em]
$ $
\begin{array}{l}
\exptype{List}{String} \lessdot {\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}},
\exptype{List}{Integer} \lessdot {\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}}
\\
\hline
\type{String} \doteq \wtv{x1}, \wtv{x1} \lessdot \tv{u}, \tv{l} \lessdot \wtv{x1}, \\
\type{Integer} \doteq \wtv{x2}, \wtv{x2} \lessdot \tv{u}, \tv{l} \lessdot \wtv{x2}
\pause \\
\hline
\type{String} \lessdot \tv{u}, \tv{l} \lessdot \type{String},\\
\type{Integer} \lessdot \tv{u}, \tv{l} \lessdot \type{Integer}
\end{array}$
\end{frame}
\begin{frame}[fragile]
%Es gibt stellen wo capture conversion möglich ist und Stellen wo es nicht ist.
\frametitle{Wildcards are bound to Type}
\begin{lstlisting}
<X> List<List<X>> shuffle2D(List<List<X>> l) {...}
List<List<?>> l = ...;
shuffle2D(l); // Fehler!
\end{lstlisting}
\end{frame}
\begin{frame}[fragile]
%Es gibt stellen wo capture conversion möglich ist und Stellen wo es nicht ist.
\frametitle{Wildcards are bound to Type}
\begin{lstlisting}
<X> List<List<X>> shuffle2D(List<List<X>> l) {...}
(*@$\exptype{List}{\wctype{\rwildcard{A}}{List}{\rwildcard{A}}}$@*) l = ...;
shuffle2D(l); // Fehler!
\end{lstlisting}
\pause
\vfill
\begin{lstlisting}
<X> List<List<X>> shuffle2D(List<List<X>> l) {...}
(*@$\color{red}\wctype{\rwildcard{A}}{List}{\exptype{List}{\rwildcard{A}}}$@*) l = ...;
shuffle2D(l); // (*@\color{red}\textbf{Correct!}@*)
\end{lstlisting}
\end{frame} \end{frame}
\begin{frame}[fragile]{Wildcard Elminiation} \begin{frame}[fragile]{Wildcard Elminiation}
@ -795,7 +613,6 @@ $
\end{frame} \end{frame}
\begin{frame}[fragile]{Wildcard Elminiation} \begin{frame}[fragile]{Wildcard Elminiation}
\textit{Continue ...}\\ \textit{Continue ...}\\
@ -844,6 +661,47 @@ $
$ $
\end{frame} \end{frame}
\begin{frame}[fragile]{Capture Conversion during Unify}
\rulename{Capture} $
\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \vdash
C \cup \, \set{ \wctype{\overline{\wildcard{B}{\type{U}}{\type{L}}}}{C}{\ol{S}} \lessdotCC \type{T} } \\
\hline
\vspace*{-0.4cm}\\
\wildcardEnv \cup \overline{\wildcard{C}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{U}}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{L}}}
\vdash C \cup \, \set{
[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}] \exptype{C}{\ol{S}} \lessdot \type{T} }
\end{array}
%\quad \ol{Y} = \textit{fresh}(\ol{X})
\quad \begin{array}[c]{l}
\ol{\rwildcard{C}} \ \text{fresh}\\
%\text{fv}(\type{T}) \neq \emptyset
\end{array}
\end{array}
$\\[1em]
\begin{itemize}
\item Constraints: ${\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\wtv{x}},
{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\wtv{x}}$
\end{itemize}
\pause
\vfill
$\begin{array}{l}
{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\wtv{x}},
{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\wtv{x}}
\pause
\\
\hline
\highlight{\wildcard{X}{\tv{u}}{\tv{l}}} \vdash \exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{x}},
{\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\wtv{x}}
\pause
\\
\hline
\wildcard{X}{\tv{u}}{\tv{l}}, \highlight{\wildcard{Y}{\tv{u}}{\tv{l}}} \vdash \exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{x}},
{\exptype{List}{\highlight{\rwildcard{Y}}}} \lessdotCC \exptype{List}{\wtv{x}}
\end{array}$
\end{frame}
\begin{frame}[fragile]{Wildcard Elimination} \begin{frame}[fragile]{Wildcard Elimination}
\textit{Continue ...}\\ \textit{Continue ...}\\
$ $
@ -934,9 +792,9 @@ $\exptype{List}{\tv{x}} \lessdot {\tv{r}}, \highlight{\type{Object}} \lessdot \t
<X> List<X> concat(List<X> l1, List<X> l2){ ... } <X> List<X> concat(List<X> l1, List<X> l2){ ... }
(*@\color{red}\texttt{List<Object>}@*) someList(){ (*@\color{red}\texttt{List<Object>}@*) someList(){
if(Math.random > 0.5){ if(Math.random > 0.5){
return new List("String"); return new List(*@\color{red}\texttt{<Object>}@*)("String");
} else { } else {
return new List(42); return new List(*@\color{red}\texttt{<Object>}@*)(42);
} }
} }
@ -946,93 +804,83 @@ concat(someList(), someList())
% wieso dürfen normale Typvariablen keine freien Variablen enthalten? % wieso dürfen normale Typvariablen keine freien Variablen enthalten?
% sonst könnte man der shuffle2D Method ein % sonst könnte man der shuffle2D Method ein
\begin{frame}[fragile] % \begin{frame}[fragile]
\begin{verbatim} % \begin{verbatim}
<A> List<A> shuffle(List<A> l){...} % <A> List<A> shuffle(List<A> l){...}
<A,B> List<B> map(List<A> l, Function<A,B> f) { ... } % <A,B> List<B> map(List<A> l, Function<A,B> f) { ... }
List<List<?>> l; % List<List<?>> l;
l.map((List<?> x) -> shuffle(x)); % l.map((List<?> x) -> shuffle(x));
\end{verbatim} % \end{verbatim}
$ % $
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{x}}, % \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{x}},
\exptype{List}{\tv{x}} \lessdot \tv{r} % \exptype{List}{\tv{x}} \lessdot \tv{r}
$ % $
\end{frame} % \end{frame}
\begin{frame}[fragile]
\begin{verbatim}
List<List<?>> l;
l2 = l.map(x -> shuffle(x));
shuffle2D(l2);
\end{verbatim}
\begin{frame}[fragile]{Wildcard Placeholders}
\begin{lstlisting}
List<List<?>> l;
(*@\only<2>{\color{red}$\exptype{List}{\exptype{List}{\rwildcard{A}}}$}@*)l2 = l.map(x -> shuffle(x)(*@\only<2>{\color{red} : $\exptype{List}{\rwildcard{A}}$}@*));
shuffle2D(l2); (*@\only<2>{\color{red} // ERROR!}@*)
\end{lstlisting}
\vfill
\textbf{Constraints:}\\
$ $
\begin{array}{l} \begin{array}{l}
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{x}}, \\ \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\wtv{x}}, \\
\exptype{List}{\exptype{List}{\tv{x}}} \lessdot \tv{l2}, \\ \exptype{List}{\exptype{List}{\wtv{x}}} \lessdot \tv{l2}, \\
\tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}} \tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\wtv{y}}}
\end{array} \end{array}
$ $
\\ \pause
\vfill
\begin{itemize} \begin{itemize}
\item Falsche Lösung: \item Falsche Lösung:
$\color{red}\tv{l2} \doteq \exptype{List}{\exptype{List}{\rwildcard{A}}}$\\
% $
% \begin{array}{l}
% \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\highlight{\rwildcard{A}}}, \\
% \exptype{List}{\exptype{List}{\tv{x}}} \lessdot \exptype{List}{\exptype{List}{\tv{x}}},
% \tv{r} \lessdot \tv{l2},
% \tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
% \end{array}
% $
$ $
\begin{array}{l} \begin{array}{l}
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\highlight{\rwildcard{A}}}, \\ \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\highlight{\rwildcard{A}}}, \\
\exptype{List}{\exptype{List}{\tv{x}}} \lessdot \exptype{List}{\exptype{List}{\tv{x}}}, \exptype{List}{\exptype{List}{\tv{x}}} \lessdot \exptype{List}{\exptype{List}{\rwildcard{A}}}, \\
\tv{r} \lessdot \tv{l2}, \exptype{List}{\exptype{List}{\rwildcard{A}}} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
\tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
\end{array} \end{array}
$ $
\pause
\item Korrekte Lösung = $\emptyset$. Programm ist inkorrekt
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\begin{frame}[fragile]{Lösung durch Unify}
$
\begin{array}{l}
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{x}}, \\
\exptype{List}{\exptype{List}{\tv{x}}} \lessdot \tv{l2}, \\
\tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
\end{array}
$
\begin{itemize}
\item Falsche Lösung:
$
\begin{array}{l}
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\highlight{\rwildcard{A}}}, \\
\exptype{List}{\exptype{List}{\tv{x}}} \lessdot \exptype{List}{\exptype{List}{\tv{x}}},
\tv{r} \lessdot \tv{l2},
\tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
\end{array}
$
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\begin{verbatim}
List<List<?>> l;
l2 = l.map(x -> shuffle(x));
shuffle2D(l2);
\end{verbatim}
$
\begin{array}{l}
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{x}}, \\
\exptype{List}{\exptype{List}{\tv{x}}} \lessdot \tv{r},
\tv{r} \lessdot \tv{l2},
\tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
\end{array}
$
\end{frame}
% könnte es nicht well-formed typen geben, wenn man beliebige substitutionen zulässt?
% (brauchen wir die wildcard-TVs überhaupt? es wird eigentlich nur für den Soundness-Beweis gebraucht)
\begin{frame}[fragile]{Conclusion} \begin{frame}[fragile]{Conclusion}
\begin{itemize} \begin{itemize}
\item Wildcard creation at $\type{T} \lessdot \tv{a}$ constraints
\begin{itemize}
\item Wildcards are bound to a type
\end{itemize}
\pause
\item Eliminate Wildcards by setting upper = lower bound \item Eliminate Wildcards by setting upper = lower bound
\begin{itemize}
\item Removes Wildcards without backtracking \item Removes Wildcards without backtracking
\end{itemize} \end{itemize}
\pause
\item Capture Constraints for Capture Conversion during \unify{}
\pause
\item Wildcard Type Placeholders to keep free variables in scope
\pause
\vfill
\item \unify{} is Sound
\begin{itemize}
\pause
\item (but no completeness proof yet)
\end{itemize}
\end{itemize}
\end{frame} \end{frame}
\end{document} \end{document}