Final Version

vortrag.tex

@@ -61,8 +61,8 @@
 		xrightmargin=0pt,		% Rand rechts
 		%frame=single,			% Rahmen an
 		%frameround=ffff,
-		rulecolor=\color{darkgray},	% Rahmenfarbe
+		rulecolor=\color{black},	% Rahmenfarbe
-		fillcolor=\color{ListingBackground},
+		fillcolor=\color{black},
         showtabs=false,
         %breaklines=true,
         %breakatwhitespace=true,
@@ -230,9 +230,9 @@
 \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");
+        return new List<>("String");
     } else {
-        return new List(42);
+        return new List<>(42);
     }
 }
 \end{lstlisting}
@@ -324,134 +324,42 @@ concat(l1, l2); //Error
     $\begin{array}{l}
         \begin{array}{@{}c}
             \textit{mtype}(\texttt{concat}) = \generics{\type{X}}\ \exptype{List}{\type{X}} \to \exptype{List}{\type{X}} \to \exptype{List}{\type{X}} \\
-            \texttt{l1} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}}, \texttt{l2} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \\
+            \texttt{l1} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}}, \texttt{l2} : \wctype{\rwildcard{B}}{List}{\rwildcard{B}} \\
             \exptype{List}{\rwildcard{A}} \nless: [{\color{red}?}/\type{X}]\exptype{List}{\type{X}} \quad \quad
 \exptype{List}{\rwildcard{B}} \nless: [{\color{red}?}/\type{X}]\exptype{List}{\type{X}}
             \\
             \hline
         \vspace*{-0.3cm}\\
-            \texttt{shuffle}(\expr{l}) : {\color{red}\textbf{Errror}}
+            \texttt{concat}(\expr{l1}, \expr{l2}) : {\color{red}\textbf{Errror}}
         \end{array}
     \end{array}$
 \end{center}
 \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{lstlisting}
 (*@\only<2>{$\color{red}\tv{r}$ }@*)someList(){
     if(Math.random > 0.5){
-        return new List("String");
+        return new List<(*@\only<2>{$\color{red}\tv{x}$}@*)>("String");
     } else {
-        return new List(42);
+        return new List<(*@\only<2>{$\color{red}\tv{y}$}@*)>(42);
     }
 }
     \end{lstlisting}
     \pause
 $
-\exptype{List}{\tv{x}} \lessdot {\only<2>{\color{red}}\tv{r}}, \type{String} \lessdot \tv{x},
+\exptype{List}{\tv{x}} \lessdot {\only<2>{\color{red}}\tv{r}}, \type{String} \lessdot {\only<2>{\color{red}}\tv{x}},
-\exptype{List}{\tv{y}} \lessdot {\only<2>{\color{red}}\tv{r}}, \type{Integer} \lessdot \tv{y}
+\exptype{List}{\tv{y}} \lessdot {\only<2>{\color{red}}\tv{r}}, \type{Integer} \lessdot {\only<2>{\color{red}}\tv{y}}
 $\\[1em]
-\pause
-\rulename{General} $
-            \begin{array}[c]{l}
-              \wildcardEnv \vdash C \cup \wctype{\Delta}{C}{\ol{T}} \lessdot \ntv{a}\\
-              \hline
-              \wildcardEnv \vdash C \cup \set{\wctype{\Delta}{C}{\ol{T}} \lessdot \ntv{a},
-            \ntv{a} \doteq \wctype{\overline{\wildcard{A}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{A}}}
-            }
-            \end{array} \quad \begin{array}[c]{l}
-              \texttt{class} \ \exptype{C}{\ol{X}} \set{\ldots }\\%\triangleleft \exptype{D}{\ol{N}} \\
-            \tv{u}, \tv{l}\ \text{fresh}
-            \end{array}
-              $
-\\[1em]
-\pause
-$
-\exptype{List}{\tv{x}} \lessdot {\tv{r}}, \highlight{\tv{x} \doteq \type{String}},
-\exptype{List}{\tv{y}} \lessdot {\tv{r}}, \highlight{\tv{y} \doteq \type{Integer}}
-$
-\pause
-\\[0.5em]
-\textbf{Substitution:}
-
-$
-\exptype{List}{\highlight{\type{String}}} \lessdot \tv{r},
-\exptype{List}{\highlight{\type{Integer}}} \lessdot \tv{r}
-$
 \end{frame}
 
 \begin{frame}[fragile]{Wildcard Creation}
     \begin{lstlisting}
 (*@\only<2>{$\color{red}\tv{r}$ }@*)someList(){
     if(Math.random > 0.5){
-        return new List("String");
+        return new List<>("String");
     } else {
-        return new List(42);
+        return new List<>(42);
     }
 }
     \end{lstlisting}
@@ -526,58 +434,42 @@ $
 \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{String} \doteq \wtv{x1}, \wtv{x1} \lessdot \tv{u}, \tv{l} \lessdot \wtv{x1}, \hfill
 \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{String} \lessdot \tv{u}, \tv{l} \lessdot \type{String},\hfill
 \type{Integer} \lessdot \tv{u}, \tv{l} \lessdot \type{Integer}
 \end{array}$
 \end{frame}
 
-\begin{frame}[fragile]
+\begin{frame}[fragile]{Wildcard Eliminiation}
-    %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}
-
-\begin{frame}[fragile]{Wildcard Elminiation}
 \begin{lstlisting}
 <X> List<X> concat(List<X> l1, List<X> l2){ ... }
+
+List<? extends Object> someList(){
+    if(Math.random > 0.5){
+        return new List<>("String");
+    } else {
+        return new List<>(42);
+    }
+}
+\end{lstlisting}
+
+\end{frame}
+\begin{frame}[fragile]{Wildcard Eliminiation}
+\begin{lstlisting}
+<X> List<X> concat(List<X> l1, List<X> l2){ ... }
+
 someList(){
     if(Math.random > 0.5){
-        return new List("String");
+        return new List<>("String");
     } else {
-        return new List(42);
+        return new List<>(42);
     }
 }
 
-concat(someList(), someList())
+(*@\color{red}\texttt{concat(someList(), someList())}@*)
 \end{lstlisting}
 
 \only<1>{\textbf{Input}:}
@@ -589,8 +481,8 @@ $
 \begin{array}{l}
 \exptype{List}{String} \lessdot {\tv{r}},
 \exptype{List}{Integer} \lessdot {\tv{r}},
-\tv{r} \lessdotCC \exptype{List}{\wtv{x}},
+\highlight{\tv{r} \lessdotCC \exptype{List}{\wtv{x}},
-\tv{r} \lessdotCC \exptype{List}{\wtv{x}}
+\tv{r} \lessdotCC \exptype{List}{\wtv{x}}}
 \pause
 \\
 \hline
@@ -613,54 +505,6 @@ $
 
 \end{frame}
 
-\begin{frame}[fragile]{Wildcard Elminiation}
-
-\textit{Continue ...}\\
-$
-\begin{array}{l}
-\type{String} \lessdot \tv{u}, \tv{l} \lessdot \type{String},
-\type{Integer} \lessdot \tv{u}, \tv{l} \lessdot \type{Integer}, \\
-\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{array}$\\[1em]
-\pause
-\rulename{Super} \\
-$
-\begin{array}[c]{l}
-\wildcardEnv \vdash C \cup \wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a}\\
-\hline
-\wildcardEnv \vdash C \cup \set{ \wctype{\Delta'}{D}{[\ol{T}/\ol{X}]\ol{N}} \lessdot \tv{a}   }
-%\set{\wctype{\ol{\wtype{W}}}{D}{[\ol{X}/\ol{Y}]\ol{Z}} \lessdot \tv{a}}
-\end{array} \quad
-\begin{array}{l}
-\texttt{class} \ \exptype{C}{\ol{X}} \ \texttt{extends} \ \exptype{D}{\ol{N}} \\
-\end{array}
-$\\[1em]
-
-\textit{Subtyping:}
-\begin{lstlisting}
-class String extends Object {...}
-class Integer extends Object {...}
-\end{lstlisting}
-\pause
-$
-\begin{array}{l}
-\type{String} \lessdot \tv{u}, \tv{l} \lessdot \type{String},
-\type{Integer} \lessdot \tv{u}, \tv{l} \lessdot \type{Integer}, \\
-\hline
-\highlight{
-\type{Object}} \lessdot \tv{u}, \tv{l} \lessdot \type{String},
-\highlight{\type{Object}} \lessdot \tv{u}, \tv{l} \lessdot \type{Integer}
-\\
-\hline
-\tv{u} \doteq \type{Object}, \ldots \\
-\pause
-{\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{array}
-$
-\end{frame}
-
 \begin{frame}[fragile]{Capture Conversion during Unify}
 \rulename{Capture} $
    \begin{array}[c]{@{}ll}
@@ -698,7 +542,7 @@ $\begin{array}{l}
 \\
 \hline
 \wildcard{X}{\tv{u}}{\tv{l}}, \highlight{\wildcard{Y}{\tv{u}}{\tv{l}}} \vdash \exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{x}},
-{\exptype{List}{\highlight{\rwildcard{Y}}}} \lessdotCC \exptype{List}{\wtv{x}}
+{\exptype{List}{\highlight{\rwildcard{Y}}}} \lessdot \exptype{List}{\wtv{x}}
 \end{array}$
 \end{frame}
 
@@ -706,23 +550,7 @@ $\begin{array}{l}
 \textit{Continue ...}\\
 $
 \begin{array}{l}
-\tv{u} \doteq \type{Object}, \tv{l} \lessdot \type{String}, \tv{l} \lessdot \type{Integer}, \\
+\ldots \\
-\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
-\tv{l} \lessdot \type{String}, \tv{l} \lessdot \type{Integer}, \\
-\wctype{\wildcard{X}{\type{Object}}{\tv{l}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{x}},
-\wctype{\wildcard{X}{\type{Object}}{\tv{l}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{x}}
-\pause
-\\
-\hline
-\wildcard{X}{\tv{u}}{\tv{l}} \vdash \tv{l} \lessdot \type{String}, \tv{l} \lessdot \type{Integer}, \\
-\exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{x}},
-\wctype{\wildcard{X}{\type{Object}}{\tv{l}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{x}}
-\pause
-\\
 \hline
 \wildcard{X}{\type{Object}}{\tv{l}}, \wildcard{Y}{\type{Object}}{\tv{l}}, \vdash \tv{l} \lessdot \type{String}, \tv{l} \lessdot \type{Integer}, \\
 \exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{x}},
@@ -740,18 +568,12 @@ $
 \rwildcard{X} \doteq \rwildcard{Y}
 \end{array}$\\[0.5em]
 \pause
-\only<7>{
+\begin{itemize}
-\rulename{Normalize}
+\item Remove Wildcard by setting lower bound = upper bound
-$\begin{array}[c]{l}
+\end{itemize}
-\wildcardEnv \cup \set{\wildcard{A}{\type{U}}{\type{L}}} \vdash C \cup \set{\rwildcard{A} \doteq \type{T}}\\
+
-                     \hline
-[\type{U}/\type{A}]\wildcardEnv \vdash [\type{U}/\type{A}]
-C \cup [\type{U}/\type{A}]\set{\ntv{a} \doteq \type{T}, \type{L} \doteq \type{U}}
-\end{array}
-$
-}
-\pause
 $\begin{array}{l}
+\ldots \\
 \hline
 \tv{l} \lessdot \type{String}, \tv{l} \lessdot \type{Integer}, \highlight{\tv{l} \doteq \type{Object}}
 \implies \textbf{Error!}
@@ -761,11 +583,12 @@ $\begin{array}{l}
 \begin{frame}[fragile]{Backtracking}
 \begin{lstlisting}
 <X> List<X> concat(List<X> l1, List<X> l2){ ... }
+
 someList(){
     if(Math.random > 0.5){
-        return new List("String");
+        return new List<>("String");
     } else {
-        return new List(42);
+        return new List<>(42);
     }
 }
 
@@ -790,6 +613,7 @@ $\exptype{List}{\tv{x}} \lessdot {\tv{r}}, \highlight{\type{Object}} \lessdot \t
 \begin{frame}[fragile]{Correct Solution}
 \begin{lstlisting}
 <X> List<X> concat(List<X> l1, List<X> l2){ ... }
+
 (*@\color{red}\texttt{List<Object>}@*) someList(){
     if(Math.random > 0.5){
         return new List(*@\color{red}\texttt{<Object>}@*)("String");
@@ -828,7 +652,8 @@ concat(someList(), someList())
 $
 \begin{array}{l}
 \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\wtv{x}}, \\
-\exptype{List}{\exptype{List}{\wtv{x}}} \lessdot \tv{l2}, \\
+\exptype{List}{\wtv{x}} \lessdot \tv{z}, \\
+\exptype{List}{\tv{z}} \lessdot \tv{l2}, \\
 \tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\wtv{y}}}
 \end{array}
 $