Add TODO. Fix introduction example (WIP). Add explanation to UNify (WIP)

introduction.tex

@@ -123,9 +123,9 @@ class Example {
     \begin{lstlisting}[style=fgj]
 genList() {
     if( ... ) {
-        return new List<String>();
+        return new List().add(1);
     } else {
-        return new List<Integer>();
+        return new List().add("String");
     }
 }
     \end{lstlisting}
@@ -137,9 +137,9 @@ genList() {
     \begin{lstlisting}[style=tfgj]
 List<?> genList() {
     if( ... ) {
-        return new List<String>();
+        return new Box<Integer>(1);
     } else {
-        return new List<Integer>();
+        return new Box<String>("String");
     }
 }
     \end{lstlisting}

martin.bib

@@ -67,12 +67,22 @@ keywords = {subtyping, type inference, principal types}
     year={1965}}
 
 @article{MM82,
-    author={A. Martelli and U. Montanari},
+author = {Martelli, Alberto and Montanari, Ugo},
-    title={An Efficient Unification Algorithm},
+title = {An Efficient Unification Algorithm},
-    journal={ACM Transactions on Programming Languages and Systems},
+year = {1982},
-    volume={4},
+issue_date = {April 1982},
-    pages={258-282},
+publisher = {Association for Computing Machinery},
-    year={1982}}
+address = {New York, NY, USA},
+volume = {4},
+number = {2},
+issn = {0164-0925},
+url = {https://doi.org/10.1145/357162.357169},
+doi = {10.1145/357162.357169},
+journal = {ACM Trans. Program. Lang. Syst.},
+month = {apr},
+pages = {258–282},
+numpages = {25}
+}
 
 @InProceedings{Plue07_3,
   author = 	 {Martin Pl{\"u}micke},

todo-11.4.24

@@ -0,0 +1,15 @@
+- Eingangsbeispiel
+- Motivation (gibt es ein nicht funktionierendes Beispiel für lokale Typinferenz was mit unserem algorithmus funktioniert)
+- Zeile 52 steht LetFJ, muss eingeführt werden
+	- Operatoren motivieren. Denotable expressable types
+	- Java macht es anders. es gibt keine let statements
+	- In java kann ich die lets nur in bestimmter Weise einbauen. Die ANF Transformation macht aus TamedFJ das letFJ
+
+- constraints nicht zu früh erwähnen
+	- informal erwähnen. um capture conversion zu behandeln braucht es neue constraints
+	- ganz vorne auflisten. Capture Conversion ist eine Neuheit
+
+Figure 4 und 5 in ein Bild
+ - man könnte die syntax unterlegen und sagen die unterlegten elemente fallen weg
+ - in beiden Syntaxen die Methodendeklarationen sollen gleich sein. beides return benutzen
+	- anschließend bei der convertierung zu TamedFJ nur die Änderungen beschreiben (Änderungen nur in Expressions)

unify.tex

@@ -1153,6 +1153,164 @@ $
 \label{fig:generation-rules}
 \end{figure}
 
+\begin{figure}
+
+\begin{center}
+  \fbox{
+  \begin{tabular}[t]{l@{~}l}
+    \rulename{Equals} %TODO
+    & $
+    \begin{array}[c]{l}
+        \wildcardEnv \vdash C \cup \, \set{ \wcNtype{\Delta}{N} \doteq \wcNtype{\Delta'}{N'} } \\ 
+        \hline
+        \vspace*{-0.4cm}\\
+        \wildcardEnv \vdash C \cup \, 
+        \set{
+          \wcNtype{\Delta}{N} \lessdot \wcNtype{\Delta'}{N'}, \wcNtype{\Delta'}{N'} \lessdot \wcNtype{\Delta}{N}
+        }
+    \end{array} %\quad |\Delta| = |\Delta'|
+    % \quad \text{fv}(\type{N}) = \text{fv}(\type{N'}) = \emptyset 
+    $
+     \\\\
+     \ruleReduceWC{}
+     & 
+     $
+     \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}},
+       \ol{\wtv{a}} \lessdot [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{U}, [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{L} \lessdot \ol{\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}
+     $
+   \\\\
+   \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}
+   $
+ \\\\
+   \rulename{Prepare} %The lessdotCC constraint only ensures that the left side looses its wildcardEnvironment.
+   %It does not ensure that the left side doesn't contain free variables. If you want to ensure that you have to give the left side a normal placeholder
+   & 
+   $
+   \begin{array}[c]{@{}ll}
+   \begin{array}[c]{l}
+       \wildcardEnv \vdash 
+       C \cup \, \set{ \type{N} \lessdot \type{N'} } \\ 
+     \hline
+     \vspace*{-0.4cm}\\
+       \wildcardEnv \vdash 
+       C \cup \, \set{ \type{N} \lessdotCC \type{N'} } \\ 
+   \end{array}
+    %\quad \ol{Y} = \textit{fresh}(\ol{X})
+    \quad \begin{array}[c]{l}
+     \text{fv}(\type{N'}) \subseteq \Delta_{in} \\
+     \text{wtv}(\type{N'}) = \emptyset
+     \end{array}
+   \end{array}
+   $
+\end{tabular}}
+\end{center}
+  \caption{Type Reduction rules}\label{fig:reductionRules}
+\end{figure}
+
+\subsection{Explanation}
+Our \unify{} process uses a similar concept to the standard unification by Martelli and Montanari \cite{MM82},
+consisting of terms, relations and variables.
+Instead of terms we have types of the form $\exptype{C}{\ol{T}}$ and
+the variables are called type placeholders.
+The input consist out of subtype relations.
+The goal is to find a solution (an unifier) which is a substitution for type placeholders
+which satisfies all input subtype constraints.
+
+Types are reduced until they %either reach a discrepancy like $\type{String} \doteq \type{Integer}$
+reach a form like $\tv{a} \doteq \type{T}$.
+Afterwards \unify{} substitutes type placehodler $\tv{a}$ with $\type{T}$.
+This is done until a substitution for all type placehodlers and therefore a valid solution is reached.
+The reduction and substitutions are done in the first step of the algorithm. 
+
+%The type reduction is done by the rules in figure \ref{fig:reductionRules}
+
+\unify{} cannot reduce constraints without checking a few prerequisites.
+Take the constraint $\wctype{\rwildcard{X}}{C}{\rwildcard{X}} \lessdot \exptype{C}{\wtv{a}}$ for example.
+If we apply a reduction here we get $\rwildcard{X} \doteq \wtv{a}$.
+The resulting $\sigma(\wtv{a}) = \rwildcard{X}$ seems like a correct substitution,
+but by S-Exists $\wctype{\rwildcard{X}}{C}{\rwildcard{X}} \nless: \exptype{C}{\rwildcard{X}}$.
+Reason: Free variables on the right side of a subtype relations are not allowed to show up as bound variables on the left side.
+$\rwildcard{X}$ in this case.
+Therefore the \rulename{Reduce} rule only reduces constraints where the left side does not declare any wildcards.
+But if the right side neither contains wildcard type placeholders nor free variables the constraint can be reduced anyways.
+The \rulename{Prepare} rule then converts this constraint to a capture constraint.
+Afterwards the \rulename{Capture} rule removes the wildcard declarations on the left side an the constraint can be reduced.
+
+\textit{Example} of the type reduction rules in figure \ref{fig:reductionRules} with the input 
+$\wctype{\rwildcard{X}}{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \rwildcard{X}} \lessdot \exptype{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \wtv{a}}$
+The first step is the \rulename{Capture} rule.
+%The right side of the constraint does not contain any free variables.
+$\begin{array}{c}
+\wctype{\rwildcard{X}}{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \rwildcard{X}} \lessdot \exptype{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \wtv{a}}\\
+\hline %Capture
+\wctype{\rwildcard{X}}{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \rwildcard{X}} \lessdotCC \exptype{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \wtv{a}}
+\end{array}$
+
+\begin{NiceTabular}{l}
+  $\ \wctype{\rwildcard{X}}{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \rwildcard{X}} \lessdot \exptype{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \wtv{a}}$ \\
+  $\nextdeduction{
+    \wctype{\rwildcard{X}}{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \rwildcard{X}} \lessdotCC \exptype{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \wtv{a}}
+  } $\\
+  $\nextdeduction{
+    \rwildcard{X} \vdash \exptype{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \rwildcard{X}} \lessdot \exptype{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \wtv{a}}
+  } $\\
+  $\nextdeduction{
+    \rwildcard{X} \vdash \wctype{\rwildcard{Y}}{List}{\rwildcard{Y}} \doteq \wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \rwildcard{X} \doteq \wtv{a}
+  } $\\
+  $\nextdeduction{
+      \wildcard{X}{Object}{\type{String}} \vdash
+      \type{String} \lessdot \mathcolorbox{addition}{\type{String}}, \tv{a} \doteq \rwildcard{X}
+  } $\\
+  $\nextdeduction{
+      \wildcard{X}{Object}{\type{String}} \vdash
+      \cancel{\type{String} \lessdot \rwildcard{X}}, \tv{a} \doteq \rwildcard{X}
+  } $\\
+      \CodeAfter
+      \begin{tikzpicture}
+          \node [right] at (2-|last) { \colorbox{white}{\rulename{Prepare}} } ; 
+          \node [right] at (3-|last) { \colorbox{white}{\rulename{Capture}} } ; 
+          \node [right] at (4-|last) { \colorbox{white}{\rulename{Reduce}} } ; 
+          \node [right] at (5-|last) { \colorbox{white}{\rulename{Equals}} } ; 
+          \node [right] at (6-|last) { \colorbox{white}{\rulename{Erase}} } ; 
+      \end{tikzpicture}
+  \end{NiceTabular}
+
 \subsection{Capture Conversion during Unification}
 The \unify{} algorithm applies a capture conversion when needed.
 A constraint of the form $\wcNtype{\Delta'}{N} \lessdot \type{T}$,