Added slides10
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slides/ThProgUE10.pdf
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% ..............................................................................
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% Demo of the fau-beamer template.
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% Copyright 2022 by Tim Roith <tim.roith@fau.de>
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\usepackage[institute=Tech,
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% ========================================================================================
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\title[fau-beamer]{Theorie der Programmierung}
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\subtitle{Übung 10 - System F}
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\author[L. Vatthauer]{
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Leon Vatthauer}
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%\author[T. Roith]{
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%Tim Roith\thanks{Friedrich-Alexander Universität Erlangen-Nürnberg, Department Mathematik}\and%
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%}
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\usepackage[useregional]{datetime2}
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\date{\DTMdisplaydate{2023}{7}{3}{-1}}
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% ================================================
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% The main document
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% ------------------------------------------------
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\begin{document}
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% Title page
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\begin{frame}[t, titleimage]{-}
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\titlepage%
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\newcommand{\ceil}[1]{\lceil {#1} \rceil}
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\newcommand{\typing}{
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\begin{block}{Typisierung}
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Wir lesen $\Gamma \vdash t : \alpha$ als „im Kontext $\Gamma$ hat der Term $t$ den Typ $\alpha$“ und definieren diese Relation wie folgt:
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\[
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\begin{array}{c c}
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\infer* [left=\text{(Ax)}, right=\text{($x : \alpha \in \Gamma$)}]{\;} {\Gamma \vdash x : \alpha} & \infer* [left=\text{($\rightarrow_i$)}] {\Gamma[x\mapsto \alpha] \vdash t : \beta} {\Gamma \vdash \lambda x.t : \alpha \rightarrow \beta}\\
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\\
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\multicolumn{2}{c}{
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\infer* [left=\text{($\rightarrow_e$)}] {\Gamma \vdash t : \alpha \rightarrow \beta \\ \Gamma \vdash s : \alpha} {\Gamma \vdash t\;s : \beta}
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}
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\end{array}
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\]
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\end{block}
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}
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\AtBeginSection{}
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\section{System F}
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\begin{frame}[t, fragile]{System F}
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System F ist eine Erweiterung des einfach getypten $\lambda$-Kalküls um Polymorphie. Wir erweitern die Typgrammatik:
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\[
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\alpha, \beta ::= a\;\vert\; \alpha \rightarrow \beta \;\vert\; \forall a. \alpha \qquad (a \in \mathbf{V})
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\]
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Die Typisierungsregeln sind:
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\[
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\begin{array}{c c c}
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\infer* [left=\text{(Ax)}, right=\text{($x : \alpha \in \Gamma$)}]{\;} {\Gamma \vdash x : \alpha} & \infer* [left=\text{($\rightarrow_i$)}] {\Gamma[x\mapsto \alpha] \vdash t : \beta} {\Gamma \vdash \lambda x.t : \alpha \rightarrow \beta}
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&\infer* [left=($\forall_i$), right=$a \not\in FV(\Gamma)$] {\Gamma \vdash s : \alpha} {\Gamma \vdash s : \forall a. \alpha}
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\\\\
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\multicolumn{2}{c}{
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\infer* [left=\text{($\rightarrow_e$)}] {\Gamma \vdash t : \alpha \rightarrow \beta \\ \Gamma \vdash s : \alpha} {\Gamma \vdash t\;s : \beta}
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}
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&\infer* [left=($\forall_e$)] {\Gamma \vdash s : \forall a.\alpha} {\Gamma \vdash s : (\alpha [\beta / a])}
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\end{array}
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\]
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\end{frame}
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% Introduction
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\section{Aufgabe 1 - Produkte in System F (a la Curry)}
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\begin{frame}[t, fragile]{Aufgabe 1}{Produkte in System F (à la Curry)}
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Wir kodieren das kartesische Produkt der Typen $a$ und $b$ in System F unter Verwendung des Typs $(a \times b) := \forall r. (a \rightarrow b \rightarrow r) \rightarrow r$
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\vfill
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\begin{enumerate}
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\item
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Für diese Kodierung ist die ``Konstruktions-Funktion'' $pair$, welche aus zwei Elementen der Typen $a$ bzw. $b$ ein Paar des Typs $(a \times b)$ konstruiert, wie folgt definiert:
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\[
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pair = \lambda x\;y . (\lambda f.\;f\;x\;y)
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\]
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Zeigen Sie, dass $\vdash pair : \forall a\;b.\; a \rightarrow b \rightarrow (a \times b)$ in System F gilt.
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\pause
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\\\;\\
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\item
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Geben Sie zu jeder der folgenden Funktionssignaturen eine Implementierung der jeweiligen Funktion (d.h. einen $\lambda$-Term) an und zeigen Sie, dass Ihre Implementierung den benötigten Typ hat:
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\begin{enumerate}
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\item[(a)] $fst : \forall a\;b.\;(a \times b) \rightarrow a$
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\item[(b)] $snd : \forall a\;b.\;(a \times b) \rightarrow b$
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\end{enumerate}
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\end{enumerate}
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\end{frame}
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\begin{frame}[t, fragile]{Aufgabe 1}{Produkte in System F (à la Curry)}
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Wir kodieren das kartesische Produkt der Typen $a$ und $b$ in System F unter Verwendung des Typs $(a \times b) := \forall r. (a \rightarrow b \rightarrow r) \rightarrow r$
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\vfill
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\begin{enumerate}
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\item[3.]
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Schreiben Sie unter Verwendung der obigen Funktionen eine weitere Funktion
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\[
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swap : \forall a\;b. (a \times b) \rightarrow (b \times a)
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\]
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und zeigen Sie, dass sie den korrekten Typ hat. Finden Sie also einen $\lambda$-Term $s$, so dass
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$\Gamma\vdash s : \forall a\;b. (a \times b) \rightarrow (b \times a)$, wobei
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\begin{align*}
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\Gamma := \{ &pair : \forall a\;b. a \rightarrow b \rightarrow (a \times b),
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\\&fst : \forall a\;b.\; (a \times b) \rightarrow a,
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\\&snd : \forall a\;b.\;(a \times b) \rightarrow b \}
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\end{align*}
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\end{enumerate}
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\end{frame}
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\begin{frame}[t, fragile]{Aufgabe 2}{Listen in System F (à la Curry)}
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Listen können in System F unter Verwendung des folgenden Typs kodiert werden:
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\begin{lstlisting}[keywords=List]
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List a := $\forall r.\; r\rightarrow (a \rightarrow r \rightarrow r) \rightarrow r$
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\end{lstlisting}
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In diesem Fall ergeben sich die folgenden ``Konstruktor-Funktionen'':
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\begin{lstlisting}
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nil = $\lambda$ u f. u
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cons = $\lambda$x l.$\lambda$ u f. f x (l u f)
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\end{lstlisting}
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Für einen gegebenen (und durch \texttt{nil} und \texttt{cons} konstruierten) Term \texttt{t} des Typs \lstinline[keywords=List]|List a|
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verhält sich der Term '\lstinline|t u ($\lambda$x l. s)|' also genau so wie eine Funktion \texttt{f} für die für alle x (des Typs a) und alle l (des Typs \lstinline[keywords=List]|List a|) gilt:
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\begin{alignat*}{2}
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&f\;nil &&\rightarrow_{\beta\delta}^* u\\
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&f\;(cons\;x\;nil) &&\rightarrow_{\beta\delta}^* s[l \mapsto f\;l]
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\end{alignat*}
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\vfill
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\begin{enumerate}
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\item Zeigen Sie, dass:
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\begin{itemize}
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\item[(a)] \lstinline|$\vdash$ nil : $\forall$ a. List a |
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\item[(b)] \lstinline|$\vdash$ cons : $\forall$ a.a $\rightarrow$ List a $\rightarrow$ List a|
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\end{itemize}
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\end{enumerate}
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\end{frame}
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\begin{frame}[t, fragile]{Aufgabe 2}{Listen in System F (à la Curry)}
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Listen können in System F unter Verwendung des folgenden Typs kodiert werden:
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\begin{lstlisting}[keywords=List]
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List a := $\forall r.\; r\rightarrow (a \rightarrow r \rightarrow r) \rightarrow r$
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\end{lstlisting}
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In diesem Fall ergeben sich die folgenden ``Konstruktor-Funktionen'':
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\begin{lstlisting}
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nil = $\lambda$ u f. u
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cons = $\lambda$x l.$\lambda$ u f. f x (l u f)
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\end{lstlisting}
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\vfill
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\begin{enumerate}
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\item[2.] Schreiben Sie eine Funktion \texttt{length}, welche die Länge einer Liste berechnet. Es soll gelten:
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\begin{alignat*}{2}
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&\texttt{length nil} &&\rightarrow_{\beta\delta}^* \texttt{zero}\\
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&\texttt{length (cons x l)} &&\rightarrow_{\beta\delta}^* \texttt{succ (length l)}
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\end{alignat*}
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Zeigen Sie, dass \lstinline|$\Gamma_0 \vdash$ length : $\forall$ a. List a $\rightarrow$ $\mathbb{N}$|, wobei $\Gamma_0 := \{\texttt{zero} : \mathbb{N}, succ : \mathbb{N} \rightarrow \mathbb{N}\}$.
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\end{enumerate}
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\end{frame}
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\end{document}
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