From dc416791a36d9b8b20f28e609eee100316a6a95f Mon Sep 17 00:00:00 2001 From: reijix Date: Mon, 19 Jun 2023 12:17:19 +0200 Subject: [PATCH] uebung 08 final --- .gitignore | 2 - slides/ThProgUE4.pdf | Bin 0 -> 749861 bytes slides/ThProgUE5.pdf | Bin 0 -> 786799 bytes slides/ThProgUE6.pdf | Bin 0 -> 745000 bytes slides/ThProgUE7.pdf | Bin 0 -> 725282 bytes slides/ThProgUE8.pdf | Bin 0 -> 756779 bytes template-art/FAU/FAUKennung.pdf | Bin 0 -> 226760 bytes template-art/FAU/FAUKennungWhite.pdf | Bin 0 -> 6804 bytes template-art/FAUWortmarkeBlau.pdf | Bin 0 -> 1987 bytes template-art/FAUWortmarkeWhite.pdf | 1186 ++++++++++++++++++++++++ template-art/Med/MedKennung.pdf | Bin 0 -> 7178 bytes template-art/Med/MedKennungWhite.pdf | Bin 0 -> 7149 bytes template-art/Nat/NatKennung.pdf | Bin 0 -> 8709 bytes template-art/Nat/NatKennungWhite.pdf | Bin 0 -> 8467 bytes template-art/Phil/PhilKennung.pdf | Bin 0 -> 10302 bytes template-art/Phil/PhilKennungWhite.pdf | Bin 0 -> 10275 bytes template-art/RW/RWKennung.pdf | Bin 0 -> 10639 bytes template-art/RW/RWKennungWhite.pdf | Bin 0 -> 10605 bytes template-art/RW/RechtKennung.pdf | Bin 0 -> 8487 bytes template-art/RW/RechtKennungWhite.pdf | Bin 0 -> 8450 bytes template-art/RW/WiSoENKennung.pdf | Bin 0 -> 9334 bytes template-art/RW/WiSoENKennungWhite.pdf | Bin 0 -> 9300 bytes template-art/RW/WiSoKennung.pdf | Bin 0 -> 11098 bytes template-art/RW/WiSoKennungWhite.pdf | Bin 0 -> 11060 bytes template-art/Tech/TechKennung.pdf | Bin 0 -> 6987 bytes template-art/Tech/TechKennungWhite.pdf | Bin 0 -> 6959 bytes slides08.tex => texfiles/slides08.tex | 125 ++- 27 files changed, 1284 insertions(+), 29 deletions(-) create mode 100755 slides/ThProgUE4.pdf create mode 100644 slides/ThProgUE5.pdf create mode 100644 slides/ThProgUE6.pdf create mode 100644 slides/ThProgUE7.pdf create mode 100644 slides/ThProgUE8.pdf create mode 100755 template-art/FAU/FAUKennung.pdf create mode 100755 template-art/FAU/FAUKennungWhite.pdf create mode 100755 template-art/FAUWortmarkeBlau.pdf create mode 100755 template-art/FAUWortmarkeWhite.pdf create mode 100755 template-art/Med/MedKennung.pdf create mode 100755 template-art/Med/MedKennungWhite.pdf create mode 100755 template-art/Nat/NatKennung.pdf create mode 100755 template-art/Nat/NatKennungWhite.pdf create mode 100755 template-art/Phil/PhilKennung.pdf create mode 100755 template-art/Phil/PhilKennungWhite.pdf create mode 100755 template-art/RW/RWKennung.pdf create mode 100755 template-art/RW/RWKennungWhite.pdf create mode 100755 template-art/RW/RechtKennung.pdf create mode 100755 template-art/RW/RechtKennungWhite.pdf create mode 100755 template-art/RW/WiSoENKennung.pdf create mode 100755 template-art/RW/WiSoENKennungWhite.pdf create mode 100755 template-art/RW/WiSoKennung.pdf create mode 100755 template-art/RW/WiSoKennungWhite.pdf create mode 100755 template-art/Tech/TechKennung.pdf create mode 100755 template-art/Tech/TechKennungWhite.pdf rename slides08.tex => 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zz{%?g#jPPYDFuOu*`FOEC&UlNCl%izx7rnKbW6Ng>vkWj(B!|A@Bu>Vj0wQbxv zX`CS#&ZxNTZ+eLRcGCsnseVl-0d+ZXh`2aJ0_Oru3Mwuk41w}PAcFr&jr6}#^H*-w z>`->@2@ tm#;ZvfKX{{tlgcW(dy literal 0 HcmV?d00001 diff --git a/slides08.tex b/texfiles/slides08.tex similarity index 79% rename from slides08.tex rename to texfiles/slides08.tex index 1478463..88cfd7d 100644 --- a/slides08.tex +++ b/texfiles/slides08.tex @@ -227,7 +227,7 @@ Die neue Grammatik für Typen lautet: Natürlich müssen wir damit auch neue Konstrukte einführen, die zu diesen Typkonstruktoren gehören. Die Termsprache wird also erweitert auf die Grammatik \[ -t,s ::= x \;\vert\; ts \;\vert\; \lambda x.t \;\vert\; {t, s} \;\vert\; fst\;t \;\vert\; snd\;t +t,s ::= x \;\vert\; ts \;\vert\; \lambda x.t \;\vert\; \{t, s\} \;\vert\; fst\;t \;\vert\; snd\;t \] \;\\ Geben Sie die zusätzlich benötigten Typisierungsregeln an und erweitern Sie ebenso die Auswertungsrelation um Grundreduktion für Paare. @@ -243,7 +243,7 @@ Geben Sie die zusätzlich benötigten Typisierungsregeln an und erweitern Sie eb Wir behaupten nun, dass diese Erweiterung uns auch mehr Ausdrucksstärke in der Logik auf der anderen Seite des Curry-Howard-Isomorphismus einbringt. Genauer gesagt, dass wir das folgende Fragment der intuitionistischen propositionalen Logik erhalten: \[ -\phi, \psi ::= a \;\vert\; \phi \rightarrow \psi \;\vert\; \phi \rightarrow \psi +\phi, \psi ::= a \;\vert\; \phi \rightarrow \psi \;\vert\; \phi \land \psi \] wobei $a$ propositionale Variablen sind. Als Deduktionssystem verwenden wir den Sequentenkalkül minimaler Logik mit folgenden zusätzlichen Regeln: \[ @@ -254,16 +254,57 @@ wobei $a$ propositionale Variablen sind. Als Deduktionssystem verwenden wir den Sei $\overline{\phi}$ der Typ, der entsteht, wenn man in $\phi$ alle $\land$ durch $\times$ ersetzt. Beweisen Sie: Der Sequent $\vdash \phi$ ist genau dann herleitbar, wenn der Typ $\overline{\phi}$ inhabited ist. \end{frame} -\begin{frame}[t, fragile]{Aufgabe 1.2}{Der Curry-Howard-Isomorphismus} + +\begin{frame}[t, fragile]{Aufgabe 1.2}{Überblick Sequenten- und $\lambda$-Kalkül} +\begin{minipage}{.44\textwidth} +\centering\textbf{Sequentenkalkül} +\\mit $\Gamma$ als Formelmenge $\{\phi_1, \ldots, \phi_n\}$ +\[ +\begin{array}{c c} + \multicolumn{2}{c}{\infer* [left=(AX), right=$\phi \in \Gamma$]{\;} {\Gamma \vdash \phi}} + \\\;\\ + \infer* [left=($\rightarrow_I$)] {\Gamma, \phi \vdash \psi} {\Gamma \vdash \phi \rightarrow \psi} & \infer* [left=($\land_I$)] {\Gamma \vdash \phi \\ \Gamma \vdash \psi} {\Gamma \vdash \phi \land \psi} + \\\;\\ + \multicolumn{2}{c}{\infer* [left=($\rightarrow_E$)] {\Gamma \vdash \phi \rightarrow \psi \\ \Gamma \vdash \phi} {\Gamma \vdash \psi}} + \\\;\\ + \infer* [left=($\land_{E1}$)] {\Gamma \vdash \phi \land \psi} {\Gamma \vdash \phi} & \infer* [left=($\land_{E2}$)] {\Gamma \vdash \phi \land \psi} {\Gamma \vdash \psi} +\end{array} +\] +\end{minipage} +\begin{minipage}{.54\textwidth} + \centering\textbf{Einfach getypter $\lambda$-Kalkül} + \\mit $\Gamma$ als Kontext $\{x_1 : \alpha_1, \ldots, x_n : \alpha_n\}$ + \[ + \begin{array}{c c} + \multicolumn{2}{c}{\infer* [left=(Ax), right=$x : \alpha \in \Gamma$]{\;} {\Gamma \vdash x : \alpha}} + \\\;\\ + \infer* [left=($\rightarrow_i$)] {\Gamma[x \mapsto \alpha] \vdash t : \beta} {\Gamma \vdash \lambda x.t : \alpha \rightarrow \beta} & \infer* [left=($prod$)] {\Gamma \vdash t : \tau \\ \Gamma \vdash s : \sigma} {\Gamma \vdash \{t,s\} : \tau \times \sigma} + \\\;\\ + \multicolumn{2}{c}{\infer* [left=($\rightarrow_e$)] {\Gamma \vdash t : \alpha \rightarrow \beta \\ \Gamma \vdash s : \alpha} {\Gamma \vdash ts : \beta}} + \\\;\\ + \infer* [left=($proj_1$)] {\Gamma \vdash p : \tau \times \sigma} {\Gamma \vdash fst\;p : \tau} & \infer* [left=($proj_2$)] {\Gamma \vdash p : \tau \times \sigma} {\Gamma \vdash snd\;p : \sigma} + \end{array} + \] + \end{minipage} +\end{frame} + + +\begin{frame}[t, fragile]{Aufgabe 1.3}{Der Curry-Howard-Isomorphismus} Benutzen Sie die eben hergestellte Korrespondenz, um zu zeigen, dass folgende Formel eine Tautologie ist: \[ (p \land q) \rightarrow r \rightarrow ((r \land p) \land q) \] +\vfill +\begin{block}{Erweiterte Termgrammatik} + \[ + t,s ::= x \;\vert\; ts \;\vert\; \lambda x.t \;\vert\; \{t, s\} \;\vert\; fst\;t \;\vert\; snd\;t + \] +\end{block} \end{frame} \section{Aufgabe 2 - Listen und Bäume} -\begin{frame}[t,fragile]{Aufgabe 2.1}{Listen und Bäume} +\begin{frame}[t,fragile]{Aufgabe 2}{Listen und Bäume} Wir betrachten die folgenden algebraischen Definitionen parametrischer Datentypen von Listen und Binärbäumen über einem Typparameter $a$: \begin{minipage}{.41\textwidth} @@ -276,26 +317,54 @@ data List a where \begin{minipage}{.58\textwidth} \begin{lstlisting}[language=haskell, morekeywords=Tree] data Tree a where - Leaf $\ \ $: () $\rightarrow$ Tree a + Leaf $\ \ $: a $\rightarrow$ Tree a Inner : a $\rightarrow$ Tree a $\rightarrow$ Tree a $\rightarrow$ Tree a \end{lstlisting} \end{minipage} -\;\\ -Beschreiben Sie in eigenen Worten die durch die folgenden Terme gegebenen Listen und Binärbäume mit natürlichen Zahlen bzw. zeichnen Sie diese. -\begin{multicols}{2} - \begin{itemize} - \item Nil - \item Cons 5 Nil - \item Leaf 13 - \item Inner 5 (Leaf 3) (Leaf 9) - \item Cons 5 (Cons 5 Nil) - \item Cons 1 (Cons 2 (Cons 3 4 Nil)) - \item Inner 8 (Inner 4 (Leaf 1) (Leaf 20)) (Leaf 8) - \item Inner 6 (Leaf 99) (Inner 1 (Leaf 4) (Leaf 6)) - \end{itemize} -\end{multicols} +\\\;\\\;\\ +\begin{block}{Einige Annahmen} +Wir nehmen Typ \textbf{Nat} von Konstanten $0,1,2,\ldots$ und die üblichen Grundoperationen für \textbf{Nat} als gegeben an. +Weiter nehmen wir den Typ \textbf{Bool} mit den Konstanten $True$ und $False$ sowie den Grundoperationen dazu als gegeben an. +Mit \texttt{()} bezeichnen wir den Typ $unit$; er enthält nur einen einzigen Wert, den wir ebenfalls mit \texttt{()} bezeichnen. +\\\;\\ +Weiterhin schreiben wir in Beweisen der Gleichheit zweier gegebener Terme $s$ und $t$ die Aussage $s \leftrightarrow_{\beta\delta}^* t$ abgekürzt als $s = t$. +\end{block} + \end{frame} +\begin{frame}[t,fragile]{Aufgabe 2.1}{Listen und Bäume} + Wir betrachten die folgenden algebraischen Definitionen parametrischer Datentypen von Listen und Binärbäumen über einem Typparameter $a$: + + \begin{minipage}{.41\textwidth} + \begin{lstlisting}[language=haskell] + data List a where + Nil $\ $: () $\rightarrow$ List a + Cons : a $\rightarrow$ List a $\rightarrow$ List a + \end{lstlisting} + \end{minipage} + \begin{minipage}{.58\textwidth} + \begin{lstlisting}[language=haskell, morekeywords=Tree] + data Tree a where + Leaf $\ \ $: a $\rightarrow$ Tree a + Inner : a $\rightarrow$ Tree a $\rightarrow$ Tree a $\rightarrow$ Tree a + \end{lstlisting} + \end{minipage} + \vfill + Beschreiben Sie in eigenen Worten die durch die folgenden Terme gegebenen Listen und Binärbäume mit natürlichen Zahlen bzw. zeichnen Sie diese. + \begin{multicols}{2} + \begin{itemize} + \item Nil + \item Cons 5 Nil + \item Leaf 13 + \item Inner 5 (Leaf 3) (Leaf 9) + \item Cons 5 (Cons 5 Nil) + \item Cons 1 (Cons 2 (Cons 3 4 Nil)) + \item Inner 8 (Inner 4 (Leaf 1) (Leaf 20)) (Leaf 8) + \item Inner 6 (Leaf 99) (Inner 1 (Leaf 4) (Leaf 6)) + \end{itemize} + \end{multicols} + \end{frame} + \begin{frame}[t,fragile]{Aufgabe 2.2}{Listen und Bäume} Wir betrachten die folgenden algebraischen Definitionen parametrischer Datentypen von Listen und Binärbäumen über einem Typparameter $a$: @@ -309,7 +378,7 @@ Beschreiben Sie in eigenen Worten die durch die folgenden Terme gegebenen Listen \begin{minipage}{.58\textwidth} \begin{lstlisting}[language=haskell, morekeywords=Tree] data Tree a where - Leaf $\ \ $: () $\rightarrow$ Tree a + Leaf $\ \ $: a $\rightarrow$ Tree a Inner : a $\rightarrow$ Tree a $\rightarrow$ Tree a $\rightarrow$ Tree a \end{lstlisting} \end{minipage} @@ -328,6 +397,7 @@ size (Leaf x) $\ \ \ \ \ $= 1 size (Inner x l r) = 1 + size l + size r \end{lstlisting} \end{minipage} + \vfill \begin{enumerate} \item[(a)] Welchen Typ hat \texttt{length}? Welchen hat \texttt{size}? \item[(b)] Werten Sie den Term \texttt{length (Cons 4 (Cons 89 (Cons 21 Nil)))} aus. @@ -347,7 +417,7 @@ size (Inner x l r) = 1 + size l + size r \begin{minipage}{.58\textwidth} \begin{lstlisting}[language=haskell, morekeywords=Tree] data Tree a where - Leaf $\ \ $: () $\rightarrow$ Tree a + Leaf $\ \ $: a $\rightarrow$ Tree a Inner : a $\rightarrow$ Tree a $\rightarrow$ Tree a $\rightarrow$ Tree a \end{lstlisting} \end{minipage} @@ -366,6 +436,7 @@ size (Leaf x) $\ \ \ \ \ $= 1 size (Inner x l r) = 1 + size l + size r \end{lstlisting} \end{minipage} + \vfill Schreiben Sie eine Funktion $\mathtt{element} : \mathbf{Nat} \rightarrow \mathbf{List}\;\mathbf{Nat} \rightarrow \mathbf{Bool}$, so dass \texttt{element a xs = True} wenn \texttt{a} in \texttt{xs} vorkommt, und andernfalls \texttt{element a xs = False}. \end{frame} @@ -374,13 +445,13 @@ size (Inner x l r) = 1 + size l + size r Der Typ und die Definitionen dieser Fold-Funktionen ergeben sich dabei allein aus den Typen der Konstruktoren. Beispielsweise ist die Fold-Funktion für den Datentyp $\mathbf{Nat}\;a$ aus der vorangegangenen Übung wie folgt definiert: \begin{lstlisting}[language=haskell] - foldL : c $\rightarrow$ (a $\rightarrow$ c $\rightarrow$ c) $\rightarrow$ List a $\rightarrow$ C + foldL : c $\rightarrow$ (a $\rightarrow$ c $\rightarrow$ c) $\rightarrow$ List a $\rightarrow$ c foldL n f Nil = n foldL n f (Cons x xs) = f x (foldL n f xs) \end{lstlisting} Hierbei entspricht der Typparameter $c$ einem Ergebnistyp und die beiden Argumente $n$ und $f$ den Konstruktoren $Nil$ und $Cons$, wobei die Typen der Argumente jeweils Operationen auf dem Ergebnistyp $c$ beschreiben, mit dem eine Liste $\mathbf{List}\;a$ in einen einzelnen Wert vom Typ $c$ „zusammengefaltet“ werden kann. - \\\;\\ + \vfill Werten Sie den Term \texttt{foldL n f (Cons 2 (Cons 3 (Cons 6 Nil)))} so weit es geht aus. \end{frame} @@ -389,13 +460,13 @@ size (Inner x l r) = 1 + size l + size r Der Typ und die Definitionen dieser Fold-Funktionen ergeben sich dabei allein aus den Typen der Konstruktoren. Beispielsweise ist die Fold-Funktion für den Datentyp $\mathbf{Nat}\;a$ aus der vorangegangenen Übung wie folgt definiert: \begin{lstlisting}[language=haskell] - foldL : c $\rightarrow$ (a $\rightarrow$ c $\rightarrow$ c) $\rightarrow$ List a $\rightarrow$ C + foldL : c $\rightarrow$ (a $\rightarrow$ c $\rightarrow$ c) $\rightarrow$ List a $\rightarrow$ c foldL n f Nil = n foldL n f (Cons x xs) = f x (foldL n f xs) \end{lstlisting} Hierbei entspricht der Typparameter $c$ einem Ergebnistyp und die beiden Argumente $n$ und $f$ den Konstruktoren $Nil$ und $Cons$, wobei die Typen der Argumente jeweils Operationen auf dem Ergebnistyp $c$ beschreiben, mit dem eine Liste $\mathbf{List}\;a$ in einen einzelnen Wert vom Typ $c$ „zusammengefaltet“ werden kann. - \\\;\\ + \vfill Finden Sie den Typ und die Definition der entsprechenden Fold-Funktion \texttt{foldT} des parametrischen Datentyps $\mathbf{Tree}\;a$. \end{frame} @@ -404,13 +475,13 @@ size (Inner x l r) = 1 + size l + size r Der Typ und die Definitionen dieser Fold-Funktionen ergeben sich dabei allein aus den Typen der Konstruktoren. Beispielsweise ist die Fold-Funktion für den Datentyp $\mathbf{Nat}\;a$ aus der vorangegangenen Übung wie folgt definiert: \begin{lstlisting}[language=haskell] - foldL : c $\rightarrow$ (a $\rightarrow$ c $\rightarrow$ c) $\rightarrow$ List a $\rightarrow$ C + foldL : c $\rightarrow$ (a $\rightarrow$ c $\rightarrow$ c) $\rightarrow$ List a $\rightarrow$ c foldL n f Nil = n foldL n f (Cons x xs) = f x (foldL n f xs) \end{lstlisting} Hierbei entspricht der Typparameter $c$ einem Ergebnistyp und die beiden Argumente $n$ und $f$ den Konstruktoren $Nil$ und $Cons$, wobei die Typen der Argumente jeweils Operationen auf dem Ergebnistyp $c$ beschreiben, mit dem eine Liste $\mathbf{List}\;a$ in einen einzelnen Wert vom Typ $c$ „zusammengefaltet“ werden kann. - \\\;\\ + \vfill Primitiv rekursive Funktionen auf $\mathbf{List}\;a$ können durch geeignete Instantiierung des Typparameters $c$ und der Argumente $n$ und $f$ alternativ mittels \texttt{foldL} ausgedrückt werden. Drücken Sie die Funktionen \texttt{length} und \texttt{size} jeweils als Folds über Listen und Bäumen aus. \end{frame}

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