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+++ b/merkzettel.tex @@ -1,5 +1,5 @@ \documentclass[11pt,a4paper]{scrartcl} -\usepackage[a4paper,left=1.5cm,right=3.0cm,top=2.5cm,bottom=2.5cm]{geometry} +\usepackage[a4paper,left=3cm,right=3cm,top=2.5cm,bottom=2.5cm]{geometry} \usepackage[ngerman]{babel} \usepackage{amssymb} \usepackage{amsthm} @@ -110,7 +110,9 @@ \item Subjektreduktion: Wenn $\Gamma\vdash t:\alpha$ und $t \rightarrow_\beta^* s$, dann auch $\Gamma \vdash s:\alpha$, aber nicht umgekehrt! \satz{3.38}{45} \end{itemize} \section*{Induktive Datentypen} +\subsection*{Mengenkonstruktionen} \subsection*{Mehrsortigkeit} + \subsection*{Strukturelle Induktion} \begin{itemize} \item über einsortige Datentypen \siehe{S. 63} @@ -135,8 +137,61 @@ \end{itemize} \subsection*{Kodatentypen mit Alternativen} \section*{System F} -\section*{Polymorphie} +\subsection*{Curry} +\begin{itemize} + \item Typen:\defin{5.1}{84} + \begin{align*} + \alpha, \beta \coloneqq a\ \vert\ \alpha \rightarrow \beta\ \vert\ \forall a.\alpha && (a\in V) + \end{align*} + \item Typisierung: \textbf{TODO: $\lambda \rightarrow$}, \textbf{TODO} \defin{5.1}{84} + \begin{align*} + Inhalt... + \end{align*} +\end{itemize} +\subsection*{Church-Kodierung}\siehe{S. 84} +\begin{itemize} + \item Natürliche Zahlen + \begin{align*} + \textbf{Todo} + \end{align*} + \item Paare + \begin{align*} + Inhalt... + \end{align*} + \item Summen + \begin{align*} + Inhalt... + \end{align*} + \item Listen + \begin{align*} + Inhalt... + \end{align*} +\end{itemize} \subsection*{ML-Polymorphie} +\begin{itemize} + \item Einschränkung von System F durch $\forall$ nur auf oberster Ebene sowie Mehrfachinstanziierung polymorpher Funktionen nur in $let$-Konstrukt \siehe{S. 88} + \item Typen + \begin{equation*} + \alpha, \beta \coloneqq a\ \vert\ \alpha \rightarrow \beta + \end{equation*} + \item Typschemata + \begin{align*} + S \coloneqq \forall a_1, \ldots, a_k.\alpha &&(k\geq 0) + \end{align*} + \item Terme + \begin{equation*} + t,s \coloneqq x\ \vert\ t\ s\ \vert\ \lambda x.t\ \vert\ let\ x = t\ in\ s + \end{equation*} + \item Kontexte + \begin{align*} + \Gamma &= ( x_1:S_1,\ldots, x_n :S_n )\\ + Cl(\Gamma, \alpha) &= \forall a_1,\ldots,a_k.\alpha &&\text{für $FV(\alpha)\backslash FV(\Gamma) = \lbrace a_1,\ldots,a_k \rbrace$} + \end{align*} + \item Typisierungsregeln \textbf{TODO} + \item Inversionslemma \textbf{TODO} + \item erweiterter Algorithmus W \textbf{TODO} +\end{itemize} +\section*{Reguläre Ausdrücke} \section*{Unifikationsalgorithmus (Martelli/Montanari)} \begin{align*} &S \cup \lbrace x \pteq x\rbrace &&\rightarrow S &&\text{(delete)}\\