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zi44JO%{NwVH7Y7lRKvOkbKJLdcz)Q6rjE*|XwXl}8dLV&*^BF7#RpYI#dla5FPCdS zYKG$~BZRc-UZNaHlj3~{C;SkGb4`V9|M?Sjb|10YHu*OfWZDlA*h``8Op+ul8exuQ zeV{qBMJ4Ncn8TD^mkfVZM_yS8KP=u(fA+2{;B=R@!ZAR-0Q7TBOMCd7qX2XkAvc%! zcPU0BT)8d8zAf0$?+Nid+3NQf3Auw%{J($Q-NVey!^h3a8jG7tfQwHUi=JLqT@LGi E0Iz4U+yDRo diff --git a/merkzettel.tex b/merkzettel.tex index 6ab6271..3d7a0f4 100644 --- a/merkzettel.tex +++ b/merkzettel.tex @@ -16,6 +16,7 @@ \usepackage{pdfpages} \usepackage{pgfplots} \usepackage{hyperref} +\usepackage{upgreek} \usepackage{tikz} \usetikzlibrary{positioning} \usetikzlibrary{arrows.meta} @@ -30,6 +31,7 @@ %Abkürzungen für Symbole, Reduktionen, etc. \newcommand{\app}{\ensuremath{\rightarrow_a}} \newcommand{\norm}{\ensuremath{\rightarrow_n}} +\newcommand{\pteq}{\ensuremath{\overset{\cdot}{=}}} \theoremstyle{definition} \newtheorem{definition}{Definition}[section] @@ -68,6 +70,50 @@ \item terminiert immer, falls Normalform existiert (nach Standardisierungssatz) \satz{3.17}{35} \end{itemize} \end{itemize} -\subsection*{Getypt} - +\subsection*{Einfach getypt $(\lambda \rightarrow)$} +\siehe{S. 39}\begin{itemize} + \item Church: Annotation der Variablen mit Typen, nur herleitbare Terme hinschreibbar + \item Curry: Alle Terme hinschreibbar, dann Aussondern der nicht typisierbaren + \item Typregeln: \siehe{S. 39} + \begin{itemize} + \item \textbf{TODO} + \end{itemize} + \item Typisierungsprobleme \begin{itemize} + \item Typcheck: \enquote{Gilt $\Gamma \vdash t:\alpha$?} + \item Typinferenz: \enquote{Was ist das beste $\alpha$ / Existiert $\alpha$ mit $\Gamma \vdash t:\alpha$?} + \item Type inhabitation: \enquote{Existiert $t$ mit $\Gamma t:\alpha$?} + \end{itemize} + \item Inversionslemma \lem{3.29}{41} \textbf{TODO} + \item Typinferenz \siehe{S. 41} + \begin{itemize} + \item Typsubstitution $\sigma$ ist Lösung von $\Gamma \vdash t:\alpha$, wenn $\Gamma\sigma \vdash t:\alpha\sigma$ herleitbar + \item Substitutionen: $\sigma_1$ allgemeiner als $\sigma_2 \Leftrightarrow \exists \tau. \sigma_1\tau = \sigma_2$ \siehe{GLoIn, S. 38} + \item Prinzipaltyp von $\Gamma, t$ ist $\sigma(a)$ für allgemeinste Lösung $\sigma$ von $\Gamma\vdash t:a$ ($a$ frisch) + \item Algorithmus W (Hindley/Milner) + \begin{itemize} + \item Menge $PT$ von Typgleichungen + \begin{align*} + PT(\Gamma;x;\alpha) &= \lbrace a\pteq b \vert x:\beta \in \Gamma\rbrace\\ + PT(\Gamma; \lambda x.t; \alpha) &= PT((\Gamma;x:a);t;b) \cup \lbrace a\rightarrow b \pteq \alpha \rbrace\ \text{mit $a,b$ frisch}\\ + PT(\Gamma;ts;\alpha) &= PT(\Gamma; t; a\rightarrow \alpha ) \cup PT(\Gamma;s;a)\ \text{mit $a$ frisch} + \end{align*} + \item Typinferenz des Terms $u$ mit leerem Kontext: + \begin{equation*} + \upvarepsilon \coloneqq PT(\emptyset;u;a) + \end{equation*} + $\Rightarrow$ Prinzipaltyp von $u$: $\text{mgu}(\upvarepsilon)(a)$ + \end{itemize} + \end{itemize} +\end{itemize} +\section*{Unifikationsalgorithmus (Martelli/Montanari)} +\begin{align*} +&S \cup \lbrace x \pteq x\rbrace &&\rightarrow S &&\text{(delete)}\\ +&S \cup \lbrace f(E_1, \ldots, E_n) \pteq f(D_1, \ldots, D_n)\rbrace &&\rightarrow S \cup \lbrace E_1 \pteq D_1, \ldots, E_n \pteq D_n\rbrace &&\text{(decomp)}\\ +&S \cup \lbrace f(E_1, \ldots, E_n) \pteq g(D_1, \ldots, D_k)\rbrace &&\rightarrow \bot \text{ (für $f \not = g$)} &&\text{(conflict)}\\ +&S \cup \lbrace E \pteq x \rbrace &&\rightarrow S \cup \lbrace x \pteq E \rbrace \text{ (für $E$ keine Variable)} &&\text{(orient)}\\ +&S \cup \lbrace x \pteq E \rbrace &&\rightarrow \begin{cases*} +\bot (\text{für }x \in FV(E), x\not = E) \\ +S\lbrack E/x\rbrack \cup \lbrace x \pteq E\rbrace (\text{für }x\notin FV(E), x\in FV(S)) +\end{cases*} && \text{ (occurs)/(elim)} +\end{align*} \end{document} \ No newline at end of file