Cleanup Unify. Add explanation to adopt rule and add lessdot markers

introduction.tex

@@ -287,10 +287,10 @@ $\begin{array}{c}
 \wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}, \, \type{String} \lessdotCC \wtv{a}
 \\
 \hline
-\textit{Capture Conversion:}\ \exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}, \, \type{String} \lessdot \wtv{a}
+\textit{Capture Conversion:}\ \wildcard{Y}{\type{Object}}{\type{String}} \wcSep \exptype{List}{\rwildcard{Y}} \lessdot \exptype{List}{\wtv{a}}, \, \type{String} \lessdot \wtv{a}
 \\
 \hline
-\textit{Solution:}\ \wtv{a} \doteq \rwildcard{X} \implies \exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\rwildcard{X}}, \, \type{String} \lessdot \rwildcard{X}
+\textit{Solution:}\ \wtv{a} \doteq \rwildcard{Y} \implies \wildcard{Y}{\type{Object}}{\type{String}} \wcSep \exptype{List}{\rwildcard{Y}} \lessdot \exptype{List}{\rwildcard{Y}}, \, \type{String} \lessdot \rwildcard{Y}
 \end{array}
 $
 \end{center}
@@ -304,7 +304,11 @@ which converts a constraint of the form $(\wctype{\rwildcard{X}}{C}{\rwildcard{X
 The type of \texttt{l} can be capture converted by a let statement if needed (see listing \ref{lst:addExampleLet}).
 Therefore we assign the constraint $\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}$
 which allows \unify{} to do a capture conversion to $\exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$.
-\textit{Note:} The constraint $\type{String} \lessdot \rwildcard{X}$ is satisfied because $\rwildcard{X}$ has $\type{String}$ as lower bound.
+The captured wildcard $\rwildcard{X}$ gets a fresh name and is stored in the wildcard environment of the \unify{} algorithm.
+
+
+\textit{Note:} The constraint $\type{String} \lessdot \rwildcard{Y}$ is satisfied
+because $\rwildcard{Y}$ has $\type{String}$ as lower bound.
 
 
 For the example shown in listing \ref{shuffleExample} the method call \texttt{shuffle(l2d)} creates the constraints:
@@ -696,3 +700,11 @@ But afterwards a capture conversion is applied, which can generate different typ
   \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}
+
+Wildcards are not reflexive. A box of type $\wctype{\rwildcard{X}}{Box}{\rwildcard{X}}$
+is able to hold a value of any type. It could be a $\exptype{Box}{String}$ or a $\exptype{Box}{Integer}$ etc.
+Also two of those boxes do not necessarily contain the same type.
+But there are situations where it is possible to assume that.
+For example the type $\wctype{\rwildcard{X}}{Pair}{\exptype{Box}{\rwildcard{X}}, \exptype{Box}{\rwildcard{X}}}$
+is a pair of two boxes, which always contain the same type.
+Inside the scope of the \texttt{Pair} type, the wildcard $\rwildcard{X}$ stays the same.

prolog.tex

@@ -20,6 +20,7 @@
 
 \newcommand{\tifj}{\texttt{TamedFJ}}
 
+\newcommand{\wcSep}{\vdash}
 \newcommand\mv[1]{{\tt #1}}
 \newcommand{\ol}[1]{\overline{\tt #1}}
 \newcommand{\exptype}[2]{\mathtt{#1 \texttt{<} #2 \texttt{>} }}

unify.tex

@@ -6,17 +6,6 @@
 % the algorithm only removes wildcards, never adds them
 
 \section{Unify}\label{sec:unify}
-The wildcard placeholders are used for intermediat types.
-It is not possible to create all super types of a type.
-The General rule only creates the ones expressable by Java syntax, which still are infinitly many in some cases \cite{TamingWildcards}.
-%thats not true. it can spawn X^T_T2.List<X> where T and T2 are types and we need to choose one inbetween them
-Otherwise the algorithm could generate more solutions, but they have to be filterd out afterwards, because they cannot be translated into Java.
-
-Any type can be inserted into wildcard placeholders.
-Normal placeholders have to contain types, which are well-formed under the supplied environment.
-% It is important that the algorithm also works for any subset of constraints
-%TODO: The unify algorithm can do any operation on wildcard placeholders the same as on normal ones.
-%TODO: Unify could make way more substitutions for wtvs especially in step 2
 
 \subsection{Description}
 The \unify{} algorithm tries to find a solution for a set of constraints like
@@ -82,20 +71,25 @@ We define two types as equal if they are mutual subtypes of each other.
 
 
 The equality relation on Capture constraints is not reflexive.
-$(\type{T} \lessdotCC \type{S}) \neq (\type{T} \lessdotCC \type{S})$ eventhough it's the same constraint.
+A capture constraint is never equal to another capture constraint even when structurally the same
+($\type{T} \lessdotCC \type{S} \neq \type{T} \lessdotCC \type{S}$).
+An implementation of the algorithm has to take this into account.
+All constraints are stored in a set and there are no dublicates of subtype constraints in a constraint set.
+Capture constraints however have to be stored as a list or have an unique number assigned
+so that duplicates don't get automatically discarded.
 
-Capture conversion is done during the \unify{} algorithm.
-\unify{} has to make two promises to ensure soundness of our type inference algorithm.
-Capture conversion can only be applied at capture constraints.
-Free variables are not allowed to leave their scope.
-This is ensured by type variables which are not allowed to be assigned type holding free variables.
+% \subsection{Capture Conversion}
+% % TODO: Describe Capture conversion
+% Capture conversion is done during the \unify{} algorithm.
+% \unify{} has to make two promises to ensure soundness of our type inference algorithm.
+% Capture conversion can only be applied at capture constraints.
+% Free variables are not allowed to leave their scope.
+% This is ensured by type variables which are not allowed to be assigned type holding free variables.
 
 \subsection{Algorithm}
 
 \newcommand{\gtype}[1]{\type{#1}}
 %\newcommand{\tw}[1]{\tv{#1}_?}
-The \unify{} algorithm computes the type solution.
-
 \begin{description}
   \item[Input:] An environment $\Delta'$
 and a set of constraints $C = \set{ \type{T} \lessdot \type{T}, \type{T} \doteq \type{T} \ldots}$
@@ -108,21 +102,6 @@ The input constraints must be of the following format:
     $\ntype{N}, \ntype{S}$ & $::=$ & $\wctype{\overline{\wildcard{X}{U}{L}}}{C}{\ol{T}} $ & Class Type \\
 \end{tabular}\\[0.5em]
 
-\noindent
-Additional requirements:
-\begin{itemize}
-
-  \item All types have to be well-formed: $\wcNtype{\Delta}{N} \in C \implies \Delta_{in} \vdash \wcNtype{\Delta}{N} \ \ok$
-\item Naming scheme of every wildcard environment has to be the same.
-%TODO: We need this so that wildcard substitutions get the correct name. also the Equals rule needs this condition
-%Example: 
-Although alpha renaming of wildcards inside a type is allowed by the type system the \unify{} algorithm never does it.
-Renaming wildcards leads to additional problems in the substitution rules and in the result containing substitutions with renamed wildcards.
-For the \rulename{Equals} to work properly it is adviced to name all wildcards in a specific scheme.
-For example by numbering them according to their appereance inside the type parameters
-(e.g. $\wctype{\rwildcard{1}, \rwildcard{2}}{Pair}{\rwildcard{1}, \rwildcard{2}}$).
-\end{itemize}
-
 \item[Output:]
 Set of unifiers $Uni = \set{\sigma_1, \ldots, \sigma_n}$ and an environment $\Delta$
 \end{description}
@@ -142,28 +121,18 @@ The \unify{} algorithm internally uses the following data types:
 
 The $\wtv{a}$ type variables are flagged as wildcard type variables.
 These type variables can be substituted by a wildcard or a type with free wildcard variables.
-As long as a type variable is flagged with as $\wtv{a}$ it will only be used by the subst-wc rule in step 1.
-In step 2 all of the wildcard flags are dismissed.
-The output therefore never contains these flags.
-
-\unify{} applies a capture conversion everywhere it is possible (see \rulename{Capture} rule).
-Capture conversion removes a types bounding environment $\Delta$.
-Type variables used in its type parameters are now bound to a global scope and not locally anymore.
+As long as a type variable is flagged as $\wtv{a}$ it can be used by the \rulename{Subst-WC} rule in step 1.
 
 With \texttt{C} being class names and \texttt{A} being wildcard names.
 The wildcard type $\wildcard{X}{U}{L}$ consist of an upper bound $\type{U}$, a lower bound $\type{L}$
 and a name $\mathtt{X}$.
 
-The \rulename{Normalize} rule eliminates wildcards. % TODO
+The \rulename{Tame} rule eliminates wildcards. %TODO
 This is done by setting the upper and lower bound to the same value.
-\unify{} generates wildcards with the \rulename{\generalizeRule} rule.
-It is important to generate new wildcards in a standardized fashion.
-When having two constraints $\type{T} \lessdot \tv{a}$ and $\type{T} \lessdot \tv{b}$,
-then after applying the \rulename{\generalizeRule} rule the freshly generated constraints are
-$\tv{a} \doteq \type{T'}, \tv{b} \doteq \type{T'}$.
-Both type variables get assigned the same type.
-This is necessary for the \rulename{Equals} rule to work properly.
 
+\unify{} applies a capture conversion everywhere it is possible (see \rulename{Capture} rule).
+Capture conversion removes a types bounding environment $\Delta$.
+Type variables used in its type parameters are now bound to a global scope and not locally anymore.
 
 \begin{figure}
   \begin{center}
@@ -182,7 +151,7 @@ This is necessary for the \rulename{Equals} rule to work properly.
                       \text{fv}(\type{T}) \subseteq \Delta', \, \text{wtv}(\type{T}) = \emptyset
                      \end{array}$\\
   \\
-  \rulename{Remove} &
+  \rulename{Normalize} &
                    $\begin{array}[c]{l}
 \wildcardEnv \vdash C \cup \set{\ntv{a} \doteq \type{T}}\\
                      \hline
@@ -579,14 +548,14 @@ $
         \begin{array}[c]{@{}ll}
         \begin{array}[c]{l}
           \wildcardEnv \vdash C \cup \, \set{
-           \tv{b} \lessdot \tv{a},
-\tv{a} \lessdot \type{N}, \tv{b} \lessdot \type{N'}} \\ 
+           \tv{b} \lessdot_1 \tv{a},
+\tv{a} \lessdot_2 \type{N}, \tv{b} \lessdot_3 \type{N'}} \\ 
           \hline
           \vspace*{-0.4cm}\\
           \wildcardEnv \vdash C \cup \, \set{
  \tv{b} \lessdot \type{N},
-           \tv{b} \lessdot \tv{a},
-\tv{a} \lessdot \type{N} , \tv{b} \lessdot \type{N'}
+           \tv{b} \lessdot_1 \tv{a},
+\tv{a} \lessdot_2 \type{N} , \tv{b} \lessdot_3 \type{N'}
            }
         \end{array}
         \end{array}
@@ -643,35 +612,29 @@ C \cup \set{ \tv{a} \lessdot \wctype{\Delta'}{C}{[\ol{U}/\ol{X}]\ol{S}}}
 \caption{Constraint reduce rules}\label{fig:reduce-rules}
 \end{figure}
 
-The new constraint generated by the adopt rule may be eliminated by the match rule.
-The adopt rule still needs to be applied only once per constraint.
-
-Wildcards consist out of three parts.
-A name, a scope and a upper and lower bound.
-
-% The \unify{} algorithm from \cite{plue09_1} substitutes type variables with wildcards.
-% A constraint $\wctype{\wildcard{X}{\type{Object}}{\bot}}{C}{\rwildcard{X}} \lessdot \exptype{C}{\tv{a}}$
-% has no solution.
-% Replacing the type variable $\tv{a}$ with the wildcard $\rwildcard{X}$ is not correct!
-% The wildcard $\rwildcard{X}$ cannot leave its scope and the type $\exptype{C}{\rwildcard{X}}$
-% is considered invalid.
-
-Wildcards are not reflexive. A box of type $\wctype{\rwildcard{X}}{Box}{\rwildcard{X}}$
-is able to hold a value of any type. It could be a $\exptype{Box}{String}$ or a $\exptype{Box}{Integer}$ etc.
-Also two of those boxes do not necessarily contain the same type.
-But there are situations where it is possible to assume that.
-For example the type $\wctype{\rwildcard{X}}{Pair}{\exptype{Box}{\rwildcard{X}}, \exptype{Box}{\rwildcard{X}}}$
-is a pair of two boxes, which always contain the same type.
-Inside the scope of the \texttt{Pair} type, the wildcard $\rwildcard{X}$ stays the same.
+Capture constraints are treated like regular subtype constraints.
+All transformations for subtype constraints work for capture constraints aswell.
+For clarification Subtype constraints are marked with a number.
+Subtype constraints holding the same number keep their respective type.
+Newly created subtype constraints are always regular subtype constrains unless stated otherwise.
+The \rulename{Adopt} rule for example takes multiple subtype constraints and adds a new one.
+Having the constraints
+$\ntv{a} \lessdotCC \wtv{b}, \ntv{a} \lessdot \type{String}, \wtv{b} \lessdot \type{Object}$
+would lead to
+$\wtv{b} \lessdot \type{String}, \ntv{a} \lessdotCC \wtv{b}, \ntv{a} \lessdot \type{String}, \wtv{b} \lessdotCC \type{Object}$
+after applying \rulename{Adopt}.
+The new generated constraint $\wtv{b} \lessdot \type{String}$ is a normal subtype constraint.
+The type placeholders which are annotated as wildcard placeholders also stay wildcard placeholders.
+The only rule that replaces wildcard type placeholders with regular placeholders is the \rulename{Normalize} rule.
 
+The algorithm holds two sets.
+The input constraints and a wildcard environment $\wildcardEnv{}$ keeping free variables.
 The algorithm starts with an empty wildcard environment $\wildcardEnv{}$.
-Only the reduce rule adds wildcards to that environment.
+Only the \rulename{Capture} rule adds wildcards to that environment.
 And every added wildcard gets a fresh unique name.
 This ensures the wildcard environment $\wildcardEnv{}$ never
 gets the same wildcard twice.
 
-When a new type is generated by the \unify{} algorithm it always has the form
-$\wctype{\ol{\rwildcard{A}}}{C}{\ol{\rwildcard{A}}}$.
 
 \textbf{Step 1:}
 Apply the rules depicted in the figures \ref{fig:normalizing-rules}, \ref{fig:reduce-rules} and \ref{fig:wildcard-rules} exhaustively.
@@ -1007,28 +970,8 @@ We have to consider both possibilities.
 \\[1em]
 
 \noindent
-\textbf{Step 3} 
-\textbf{(Eliminate Wildcard Variables):}
-If no more rules in step 2 are applicable \unify{} has to eliminate all wildcard variables and $\lessdotCC$ constraints by applying the \rulename{Remove} rule
-and start over at step 1.
-If $C$ does not contain any wildcard variables the algorithm proceeds with step 4.
-\begin{center}
-  \fbox{
-  \begin{tabular}[t]{l@{~}l}
-\rulename{Remove-Cons} & $
-    \begin{array}[c]{l}
-       \wildcardEnv \vdash C \cup \set{\type{S} \lessdotCC \type{T} } \\
-      \hline
-      \vspace*{-0.4cm}\\
-      \wildcardEnv \vdash C \cup \set{\type{S} \lessdot \type{T} }
-    \end{array}
-    $
-\end{tabular}}
-\end{center}
-
-\noindent
-\textbf{Step 4:}
-We apply the rules in figure \ref{fig:cleanup-rules} exhaustively and proceed with step 6.
+\textbf{Step 3:}
+We apply the rules in figure \ref{fig:cleanup-rules} exhaustively and proceed with step 4.
 
 \begin{figure}
   \begin{center}
@@ -1237,3 +1180,18 @@ But this renders additional problems:
 	\item Capture Converted variables are not allowed to leave their scope
 	\item \unify{} generates type substitution which cannot be translated to Java types.
 \end{itemize}
+
+
+\subsection{Completeness}
+It is not possible to create all super types of a type.
+The General rule only creates the ones expressable by Java syntax, which still are infinitly many in some cases \cite{TamingWildcards}.
+%thats not true. it can spawn X^T_T2.List<X> where T and T2 are types and we need to choose one inbetween them
+Otherwise the algorithm could generate more solutions, but they have to be filterd out afterwards, because they cannot be translated into Java.
+
+
+\subsection{Implementation}
+%List this under implementation details
+Every constraint that is not in solved form and is not able to be processed by any of the rules accounts as a error constraint and renders the constraint set unsolvable
+
+The new constraint generated by the adopt rule may be eliminated by the match rule.
+The adopt rule still needs to be applied only once per constraint.