Cleanup
This commit is contained in:
JanUlrich
parent
f371d8ee61
commit
b31d31a063
748
vortrag.tex
748
vortrag.tex
@ -62,7 +62,7 @@
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%frame=single, % Rahmen an
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%frameround=ffff,
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rulecolor=\color{darkgray}, % Rahmenfarbe
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fillcolor=\color{ListingBackground}
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fillcolor=\color{ListingBackground},
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showtabs=false,
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%breaklines=true,
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%breakatwhitespace=true,
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@ -213,243 +213,45 @@
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\newcommand{\rulenameAfter}[1]{\begin{array}[b]{l}\rulename{#1}\\[-0.5em] \ \end{array}}
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\begin{frame}[fragile]
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\frametitle{Java Method Call}
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\rulename{T-Call} \\[1em]
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\begin{center}
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
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\ol{v} : [\ol{T'}/\ol{X}]\ol{T}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\generics{\ol{T'}}\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
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\end{array}
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\end{array}$
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\end{center}
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\textbf{Example:}
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$\generics{\type{String}}\texttt{emptyList}() : \exptype{List}{\type{String}}$
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\\[2em]
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\rulename{TI-Call} \\[1em]
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\begin{center}
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
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\ol{v} : [\ol{T'}/\ol{X}]\ol{T}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
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\end{array}
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\end{array}$
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\end{center}
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\textbf{Example:}
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$\texttt{emptyList}() : \exptype{List}{\type{String}}$
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\end{frame}
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% \begin{frame}[fragile]
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% \frametitle{Java Method Call}
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% \rulename{M-Class} \\[1em]
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% \begin{center}
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% $\begin{array}{l}
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% \begin{array}{@{}c}
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% \generics{\ol{X}}\ \type{T} \ \texttt{m}(\ol{T\ x}) \set{ \ldots }
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% \\
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% \hline
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% \vspace*{-0.3cm}\\
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% \textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T}
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% \end{array}
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% \end{array}$
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% \end{center}
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% \textbf{Beispiele:}\\
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% \begin{verbatim}
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% X head(List<X> l) { ... }
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% \end{verbatim}
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% \begin{itemize}
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% \item
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% $\textit{mtype}(\texttt{head}) = \generics{\type{X}}\ \exptype{List}{\type{X}} \to \type{X}$
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% \end{itemize}
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% \begin{verbatim}
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% <X> List<X> emptyList() { ... }
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% \end{verbatim}
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% \begin{itemize}
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% \item
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% $\textit{mtype}(\texttt{emptyList}) = \generics{\type{X}}\ [] \to \exptype{List}{\type{X}}$
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% \end{itemize}
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% \end{frame}
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\begin{frame}[fragile]
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\frametitle{Java Method Call}
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\rulename{T-Call} \\[1em]
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\begin{center}
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
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\Delta \vdash \expr{v}, \ol{v} : [\ol{T'}/\ol{X}]\ol{T}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \expr{v}.\generics{\ol{T'}}\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
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\end{array}
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\end{array}$
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\end{center}
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\textbf{Beispiel:}\\
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\begin{verbatim}
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<X> X head(List<X> l){ ... }
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head(new List<String>());
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\end{verbatim}
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\pause
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{head}) = \generics{\ol{X}}\ \exptype{List}{\type{X}} \to \type{X} \\
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\texttt{new List<String>}() : [\type{String}/\type{X}]\exptype{List}{\type{X}}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\generics{\type{String}}\texttt{head}(\texttt{new List<String>}()) : [\type{String}/\type{X}]\type{X}
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\end{array}
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\end{array}$\\[1em]
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\pause
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% $\begin{array}{l}
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% \begin{array}{@{}c}
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% \textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
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% \texttt{new List<String>}() : \exptype{List}{\type{String}}
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% \\
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% \hline
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% \vspace*{-0.3cm}\\
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% \generics{\type{String}}\texttt{head}(\texttt{new List<String>}()) : \type{String}
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% \end{array}
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% \end{array}$
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\begin{itemize}
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\item \texttt{head(new List<String>()) : String}
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\end{itemize}
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\end{frame}
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% \begin{frame}[fragile]
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% \frametitle{Java Method Call}
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% \begin{verbatim}
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% head(head(emptyList()))
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% \end{verbatim}
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% $
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% \begin{array}[b]{c}
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% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
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% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
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% \\
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% \hline
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% \texttt{head(emptyList())} : \exptype{List}{\type{String}}
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% \end{array} \rulenameAfter{T-Call}
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% $
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% \end{frame}
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% \begin{frame}[fragile]
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% \frametitle{Java Method Call}
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% $
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% \begin{array}[b]{c}
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% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
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% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
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% \\
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% \hline
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% \generics{\exptype{List}{\exptype{List}{\type{String}}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
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% \end{array} \rulenameAfter{T-Call}
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% $
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% $
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% \begin{array}[b]{c}
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% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
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% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
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% \\
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% \hline
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% \generics{\exptype{List}{\type{String}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
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% \end{array} \rulenameAfter{T-Call}
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% $
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% \end{frame}
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% \begin{frame}[fragile]
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% \frametitle{Java Method Call}
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% $
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% \begin{array}[b]{c}
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% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
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% \quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
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% \\
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% \hline
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% \generics{\exptype{List}{\type{String}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
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% \end{array} \rulenameAfter{T-Call}
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% $
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% \end{frame}
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% \begin{frame}[fragile]
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% \frametitle{Java Method Call}
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% $
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% \begin{array}[b]{c}
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% \begin{array}[b]{c}
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% \texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}}
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% \\
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% \hline
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% \texttt{emptyList().get()} : \exptype{List}{\type{String}}
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% \end{array} \rulenameAfter{T-Call}
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% \\
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% \hline
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% \texttt{emptyList().get().get()} : \type{String}
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% \end{array} \rulenameAfter{T-Call}
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% $
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% \end{frame}
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% \begin{frame}[fragile]
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% \frametitle{Method Call with Wildcards}
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% Eigentlich kann man mit Wildcard Typen keine Methodenaufrufe durchführen.
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% Der eigentliche Typ ist nicht bekannt.
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% ? sind auch nicht reflexiv
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% Es geht dennoch, weil sich hinter einem Wildcard Typ ein konkreter Typ verbirgt.
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% Dieser muss ausgepackt werden.
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% Die entstehende freie Typvariable kann nun als Typ benutzt werden.
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% \end{frame}
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\begin{frame}[fragile]
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\frametitle{Typeinference ohne Wildcards}
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\rulename{TI-Call} \\[1em]
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\begin{center}
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
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\Delta \vdash \expr{v}, \ol{v} : [\ol{T'}/\ol{X}]\ol{T}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \expr{v}.\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
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\end{array}
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\end{array}$
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\end{center}
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\begin{verbatim}
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untypedMethod(l){
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shuffle(l);
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\begin{frame}[fragile]{Global Type Inference}
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\begin{lstlisting}
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(*@\only<1>{\color{gray}\texttt{List<? extends Object>} }@*)someList(){
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if(Math.random > 0.5){
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return new List(*@\only<1>{\color{gray}\texttt{<String>}}@*)("String");
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} else {
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return new List(*@\only<1>{\color{gray}\texttt{<Integer>}}@*)(42);
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}
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}
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\end{verbatim}
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\end{lstlisting}
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\pause
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\end{frame}
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\begin{frame}[fragile]{Wildcards as Existential Types}
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\begin{lstlisting}
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(*@\only<2>{\color{red}$\wctype{\wildcard{A}{\type{Object}}{\bot}}{List}{\rwildcard{A}}$ }@*)someList(){
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if(Math.random > 0.5){
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return new List("String");
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} else {
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return new List(42);
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}
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}
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\end{lstlisting}
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\pause
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\begin{itemize}
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\item $\lessdot $
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\item \texttt{List<? extends Object>} $\implies$ $\wctype{\wildcard{A}{\type{Object}}{\bot}}{List}{\rwildcard{A}}$
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\item \texttt{List<? super String>} $\implies$ $\wctype{\wildcard{A}{\type{Object}}{\type{String}}}{List}{\rwildcard{A}}$
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\end{itemize}
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\end{frame}
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\begin{frame}[fragile]{Capture Conversion}
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\rulename{T-Call-Wildcards} \\[1em]
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\alt<-2>{
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\begin{center}
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
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\Delta \vdash \ol{v} : \ol{\wctype{\Delta}{C}{\ol{T}}} \\
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\overline{\exptype{C}{\ol{T}}} <: [\ol{T'}/\ol{X}]\ol{T} \quad \quad [\ol{T'}/\ol{X}]\type{T} <: \type{R}
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\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}
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\\
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\hline
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\vspace*{-0.3cm}\\
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@ -457,45 +259,35 @@ $\begin{array}{l}
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\end{array}
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\end{array}$
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\end{center}
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%Wildcard Capture
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\begin{verbatim}
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<X> List<X> shuffle(List<X> l){ ... }
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List<?> l;
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shuffle(l) : List<?>
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\end{verbatim}
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%why do we need capture constraints?
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% List<List<?>> not a subtype of List<List<X>> for any X
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}{
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\begin{center}
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{shuffle}) = \generics{\type{X}}\ \exptype{List}{\type{X}} \to \exptype{List}{\type{X}} \\
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\texttt{l} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \quad \quad
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\exptype{List}{\type{A}} <: [\type{A}/\type{X}]\exptype{List}{\type{X}} \\
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\ [\type{A}/\type{X}]\exptype{List}{\type{X}} <: \wctype{\rwildcard{A}}{List}{\rwildcard{A}}
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\textit{mtype}(\texttt{shuffle}) = \generics{\type{X}}\ \exptype{List}{\type{X}} \to \exptype{List}{\type{X}} \quad \quad
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\texttt{l} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \\
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\wildcard{A}{\type{Object}}{\bot} \vdash \exptype{List}{\type{A}} <: [\type{A}/\type{X}]\exptype{List}{\type{X}} \quad \quad
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\ [\type{A}/\type{X}]\exptype{List}{\type{X}} <: \wctype{\rwildcard{B}}{List}{\rwildcard{B}}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\texttt{shuffle}(\expr{l}) : \wctype{\rwildcard{A}}{List}{\rwildcard{A}}
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\texttt{shuffle}(\expr{l}) : \wctype{\rwildcard{B}}{List}{\rwildcard{N}}
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\end{array}
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\end{array}$
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\end{center}
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\end{frame}
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\begin{frame}[fragile,t]
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%Explain why ? cannot be used as a regular type
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\begin{verbatim}
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<X> List<X> concat(List<X> l1, List<X> l2){
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return l1.addAll(l2);
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}
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\pause
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%Wildcard Capture
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\begin{lstlisting}
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<X> List<X> shuffle(List<X> l){ ... }
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List<?> l1;
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List<?> l2;
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List<? extends Object> l;
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shuffle(l) (*@\only<3>{\color{red}\texttt{: List<? extends Object>}}@*)
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\end{lstlisting}
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%why do we need capture constraints?
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% List<List<?>> not a subtype of List<List<X>> for any X
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\pause
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concat(l1, l2); //Error
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\end{verbatim}
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\end{frame}
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\begin{frame}[fragile,t]
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@ -514,137 +306,100 @@ concat(l1, l2); //Error (würde l2 an ls anfügen!)
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\end{verbatim}
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\end{frame}
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\begin{frame}[fragile]
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%Wildcard Capture
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\begin{frame}[fragile,t]
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%Explain why ? cannot be used as a regular type
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\begin{verbatim}
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<X> void shuffle2D(List<List<X>> l){
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//randomly exchange every element
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// in the two dimensional list l
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<X> List<X> concat(List<X> l1, List<X> l2){
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return l1.addAll(l2);
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}
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List<List<?>> l;
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List<?> l1;
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List<?> l2;
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shuffle2D(l); // Error
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concat(l1, l2); //Error
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\end{verbatim}
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$\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \nless: \exptype{List}{\rwildcard{A}}$
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\pause
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\vfill
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\begin{center}
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$\begin{array}{l}
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\begin{array}{@{}c}
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\textit{mtype}(\texttt{concat}) = \generics{\type{X}}\ \exptype{List}{\type{X}} \to \exptype{List}{\type{X}} \to \exptype{List}{\type{X}} \\
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\texttt{l1} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}}, \texttt{l2} : \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \\
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\exptype{List}{\rwildcard{A}} \nless: [{\color{red}?}/\type{X}]\exptype{List}{\type{X}} \quad \quad
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\exptype{List}{\rwildcard{B}} \nless: [{\color{red}?}/\type{X}]\exptype{List}{\type{X}}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\texttt{shuffle}(\expr{l}) : {\color{red}\textbf{Errror}}
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\end{array}
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\end{array}$
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\end{center}
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\end{frame}
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\begin{frame}[fragile]
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%Es gibt stellen wo capture conversion möglich ist und Stellen wo es nicht ist.
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\frametitle{Capture Constraint}
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\begin{verbatim}
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<X> List<List<X>> shuffle2D(List<? extends List<X>> l) {...}
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List<List<?>> l = ...;
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shuffle2D(l); // Fehler!
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\end{verbatim}
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\begin{itemize}
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\item hier ergibt sich ein Subtype-Constraint der keine Capture Conversion zulässt:
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\item $\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{x}}$
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\pause
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\item $\unify{}(\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{a}}) = \emptyset$
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\end{itemize}
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\end{frame}
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\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
|
||||
\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:}
|
||||
$\begin{array}[t]{c}
|
||||
\wctype{\wildcard{A}{\type{U}}{\type{L}}}{List}{\type{A}} \lessdot \exptype{List}{\wtv{x}}\\
|
||||
\hline
|
||||
\wildcard{A}{\type{U}}{\type{L}} \vdash \exptype{List}{\type{A}} \lessdot \exptype{List}{\wtv{x}}
|
||||
\\
|
||||
\hline
|
||||
\wildcard{A}{\type{U}}{\type{L}} \vdash \type{A} \doteq \wtv{x}
|
||||
\end{array}$
|
||||
\end{itemize}
|
||||
\end{frame}
|
||||
% \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}
|
||||
@ -735,17 +490,80 @@ $
|
||||
$
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[fragile]
|
||||
\begin{frame}[fragile]{Reduce}
|
||||
$
|
||||
\begin{array}{l}
|
||||
\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}}}
|
||||
\\
|
||||
\hline
|
||||
\type{String} \doteq \wtv{x1}, \type{Integer} \doteq \wtv{x2}
|
||||
\end{array}
|
||||
\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}
|
||||
|
||||
\begin{frame}[fragile]{Wildcard Elminiation}
|
||||
@ -795,7 +613,6 @@ $
|
||||
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}[fragile]{Wildcard Elminiation}
|
||||
|
||||
\textit{Continue ...}\\
|
||||
@ -844,6 +661,47 @@ $
|
||||
$
|
||||
\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}
|
||||
\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){ ... }
|
||||
(*@\color{red}\texttt{List<Object>}@*) someList(){
|
||||
if(Math.random > 0.5){
|
||||
return new List("String");
|
||||
return new List(*@\color{red}\texttt{<Object>}@*)("String");
|
||||
} 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?
|
||||
% sonst könnte man der shuffle2D Method ein
|
||||
\begin{frame}[fragile]
|
||||
\begin{verbatim}
|
||||
<A> List<A> shuffle(List<A> l){...}
|
||||
<A,B> List<B> map(List<A> l, Function<A,B> f) { ... }
|
||||
List<List<?>> l;
|
||||
l.map((List<?> x) -> shuffle(x));
|
||||
\end{verbatim}
|
||||
$
|
||||
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{x}},
|
||||
\exptype{List}{\tv{x}} \lessdot \tv{r}
|
||||
$
|
||||
\end{frame}
|
||||
\begin{frame}[fragile]
|
||||
\begin{verbatim}
|
||||
List<List<?>> l;
|
||||
l2 = l.map(x -> shuffle(x));
|
||||
shuffle2D(l2);
|
||||
\end{verbatim}
|
||||
% \begin{frame}[fragile]
|
||||
% \begin{verbatim}
|
||||
% <A> List<A> shuffle(List<A> l){...}
|
||||
% <A,B> List<B> map(List<A> l, Function<A,B> f) { ... }
|
||||
% List<List<?>> l;
|
||||
% l.map((List<?> x) -> shuffle(x));
|
||||
% \end{verbatim}
|
||||
% $
|
||||
% \wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{x}},
|
||||
% \exptype{List}{\tv{x}} \lessdot \tv{r}
|
||||
% $
|
||||
% \end{frame}
|
||||
|
||||
\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}
|
||||
\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}}}
|
||||
\wctype{\rwildcard{A}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\wtv{x}}, \\
|
||||
\exptype{List}{\exptype{List}{\wtv{x}}} \lessdot \tv{l2}, \\
|
||||
\tv{l2} \lessdotCC \exptype{List}{\exptype{List}{\wtv{y}}}
|
||||
\end{array}
|
||||
$
|
||||
|
||||
\\ \pause
|
||||
\vfill
|
||||
\begin{itemize}
|
||||
\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}
|
||||
\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}}}
|
||||
\exptype{List}{\exptype{List}{\tv{x}}} \lessdot \exptype{List}{\exptype{List}{\rwildcard{A}}}, \\
|
||||
\exptype{List}{\exptype{List}{\rwildcard{A}}} \lessdotCC \exptype{List}{\exptype{List}{\tv{y}}}
|
||||
\end{array}
|
||||
$
|
||||
\pause
|
||||
\item Korrekte Lösung = $\emptyset$. Programm ist inkorrekt
|
||||
\end{itemize}
|
||||
\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{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
|
||||
\begin{itemize}
|
||||
\item Removes Wildcards without backtracking
|
||||
\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{document}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user