Introduce normal type placeholders \ntv
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Andreas Stadelmeier
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5074c21943
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fca93d7ec6
@ -111,6 +111,7 @@
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\newcommand{\wtypestore}[3]{\ensuremath{#1 = \wtype{#2}{#3}}}
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%\newcommand{\wtype}[2]{\ensuremath{[#1\ #2]}}
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\newcommand{\wtv}[1]{\ensuremath{\tv{#1}_?}}
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\newcommand{\ntv}[1]{\ensuremath{\underline{\tv{#1}}}}
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\newcommand{\wcstore}{\ensuremath{\Delta}}
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%\newcommand{\rwildcard}[1]{\ensuremath{\mathtt{#1}_?}}
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70
unify.tex
70
unify.tex
@ -1,6 +1,18 @@
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\section{Unify}\label{sec:unify}
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The wildcard placeholders are used for intermediat types.
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It is not possible to create all super types of a type.
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The General rule only creates the ones expressable by Java syntax, which still are infinitly many in some cases \cite{TamingWildcards}.
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%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
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Otherwise the algorithm could generate more solutions, but they have to be filterd out afterwards, because they cannot be translated into Java.
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Any type can be inserted into wildcard placeholders.
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Normal placeholders have to contain types, which are well-formed under the supplied environment.
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% It is important that the algorithm also works for any subset of constraints
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%TODO: The unify algorithm can do any operation on wildcard placeholders the same as on normal ones.
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%TODO: Unify could make way more substitutions for wtvs especially in step 2
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\subsection{Description}
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The \unify{} algorithm tries to find a solution for a set of constraints like
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$\set{\exptype{List}{String} \lessdot \tv{a}, \exptype{List}{Integer} \lessdot \tv{a}}$.
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@ -137,13 +149,13 @@ This is necessary for the \rulename{Equals} rule to work properly.
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\begin{tabular}[t]{l@{~}l}
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\rulename{Subst} &
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$\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \set{\tv{a} \doteq \type{T}}\\
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\wildcardEnv \vdash C \cup \set{\ntv{a} \doteq \type{T}}\\
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\hline
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[\type{T}/\tv{a}]\wildcardEnv \vdash [\type{T}/\tv{a}]
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C \cup \set{\tv{a} \doteq \type{T}}
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C \cup \set{\ntv{a} \doteq \type{T}}
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\end{array}
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\quad \begin{array}{c}
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\tv{a} \notin \type{T} \\
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\ntv{a} \notin \type{T} \\
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\text{fv}(\type{T}) \subseteq \Delta'
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\end{array}$\\
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\\
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@ -184,7 +196,7 @@ $
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\wildcardEnv \cup \set{\wildcard{A}{U}{L}} \vdash C \cup \, \set{ \type{U} \lessdot \type{G} }
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\end{array}
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\quad \quad
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\begin{array}[c]{l}
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\begin{array}[c]{l} %TODO: can the second part be removed by adding a X.C<X> <. C<a?> constraint at method invocation?
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\wildcardEnv \cup \set{\wildcard{A}{U}{L}} \vdash C \cup \, \set{ \type{A} \lessdotCC \type{G} } \\
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\hline
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\vspace*{-0.4cm}\\
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@ -329,24 +341,24 @@ $
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The \rulename{match} rule generates fresh wildcards $\overline{\wildcard{A}{\tv{u}}{\tv{l}}}$.
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Their upper and lower bounds are fresh type variables.
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%Unify only renames the wildcards in the reduce rule
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% It's the only place where wildcards are coming into play (theres always a reduce step before a wildcard substitution is possible)
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% die wildcard variablen sollten erst am Ende ausgetauscht werden gegen normale variablen
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% das funktioniert, da die im Reduce step erstellten direkt substituiert werden
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% die anderen erlauben Capture Conversion aber nur wenn der Methodentyp und Parametertyp schon feststeht! (gleich Mächtig wie TI in Java)
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% a? <. T ->
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% T <. a? ->
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% a? =. T -> substitute!
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% bei normalen Typvariablen werden keine Wildcards substituiert
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% \begin{tcolorbox}
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% $
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% \wctype{\rwildcard{X}}{Box}{\rwildcard{X}} \lessdot \exptype{Box}{\tv{a}_?}, \\
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% \exptype{Box}{\tv{a}_?} \lessdot \wctype{\rwildcard{X}}{Box}{\rwildcard{X}}
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% $
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% \end{tcolorbox}
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\begin{figure}
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\begin{center}
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\leavevmode
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\fbox{
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\begin{tabular}[t]{l@{~}l}
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\rulename{Circle} & $
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\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \, \set{\tv{a}_1 \lessdot
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\tv{a}_2, \tv{a}_2 \lessdot \tv{a}_3, \dots, \tv{a}_n \lessdot \tv{a}_1}\\
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\hline
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\wildcardEnv \vdash C \cup \, \set{\tv{a}_1 \doteq \tv{a}_2, \tv{a}_2 \doteq \tv{a}_3, \dots , \tv{a}_n \doteq \tv{a}_1}
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\end{array} \quad n>0
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$
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\end{tabular}}
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\end{center}
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\caption{Rules for normal placeholders}\label{fig:reduce-rules}
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\end{figure}
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\begin{figure}
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\begin{center}
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@ -751,10 +763,10 @@ This builds a search tree over multiple possible solutions.
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& $
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\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \type{N},
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\tv{a} \lessdot^* \tv{b}}
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\tv{a} \lessdot \tv{b}}
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\\
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\hline
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\wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot^* \tv{b}, \tv{b} \lessdot \type{N} }
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\wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \tv{b}, \tv{b} \lessdot \type{N} }
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\end{array}
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$
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\\\\
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@ -1049,13 +1061,13 @@ Otherwise the generation rules \rulename{GenSigma} and \rulename{GenDelta} will
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\rulename{GenDelta}
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& $
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\deduction{
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\wildcardEnv \vdash C \cup \set{\tv{b} \lessdot \type{T} } \implies \Delta, \sigma
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\wildcardEnv \vdash C \cup \set{\ntv{b} \lessdot \type{T} } \implies \Delta, \sigma
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}{
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\wildcardEnv \vdash [\type{B}/\tv{b}]C \implies \Delta \cup \set{\wildcard{B}{\type{T}}{\bot}}, \sigma \cup \set{\tv{b} \to \type{B}}
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\wildcardEnv \vdash [\type{B}/\ntv{b}]C \implies \Delta \cup \set{\wildcard{B}{\type{T}}{\bot}}, \sigma \cup \set{\ntv{b} \to \type{B}}
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} \quad
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\begin{array}{l}
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\tph(\type{T}) = \emptyset, \text{fv}(\type{T}) \subseteq \Delta \cup \Delta_{in} \\
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\rwildcard{B} \ \text{fresh}, \tv{b} \notin \text{dom}(\sigma), \Delta, \Delta_{in} \vdash \type{T} \ \ok
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\rwildcard{B} \ \text{fresh}, \ntv{b} \notin \text{dom}(\sigma), \Delta, \Delta_{in} \vdash \type{T} \ \ok
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\end{array}
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$
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\\\\
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@ -1063,14 +1075,14 @@ Otherwise the generation rules \rulename{GenSigma} and \rulename{GenDelta} will
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& $
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\deduction{
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\wildcardEnv \vdash C \cup
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\set{\tv{a} \doteq \type{T} } \implies \Delta, \sigma
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\set{\ntv{a} \doteq \type{T} } \implies \Delta, \sigma
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}{
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\wildcardEnv \vdash C \implies \Delta, \sigma \cup
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\set{\tv{a} \to \type{T} }
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\set{\ntv{a} \to \type{T} }
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} \quad
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\begin{array}{l}
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\tph(\type{T}) = \emptyset \\ %,\, \text{fv}(\type{T}) \subseteq \Delta \\ % T ok implies that
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\tv{a} \notin \text{dom}(\sigma),\, \Delta, \Delta_{in} \vdash \type{T} \ \ok
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\ntv{a} \notin \text{dom}(\sigma),\, \Delta, \Delta_{in} \vdash \type{T} \ \ok
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\end{array}
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$
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\\\\
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