Typregeln, Church-Kodierung, erweiterter Alg. W
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Marco Ammon
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merkzettel.pdf
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merkzettel.pdf
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merkzettel.tex
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merkzettel.tex
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@ -11,7 +11,7 @@
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\usepackage{tikz-qtree}
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\usepackage{mathtools}
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\usepackage{latexsym}
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\usepackage{algorithmicx}
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\usepackage{bussproofs}
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\usepackage{csquotes}
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\usepackage{pdfpages}
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\usepackage{pgfplots}
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@ -34,6 +34,8 @@
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\newcommand{\app}{\ensuremath{\rightarrow_a}}
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\newcommand{\norm}{\ensuremath{\rightarrow_n}}
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\newcommand{\pteq}{\ensuremath{\overset{\cdot}{=}}}
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\newcommand{\bred}{\ensuremath{\rightarrow_\beta}}
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\newcommand{\ered}{\ensuremath{\rightarrow_\eta}}
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\theoremstyle{definition}
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\newtheorem{definition}{Definition}[section]
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@ -52,7 +54,16 @@
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\section*{$\lambda$-Kalkül}
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\subsection*{Ungetypt}
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\subsubsection*{Rekursion}
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\begin{itemize}
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\item $\beta$-Reduktion
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\begin{equation*}
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(\lambda x.t)s \bred t[s/x]
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\end{equation*}
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\item $\eta$-Reduktion
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\begin{equation*}
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\lambda x.yx \ered y
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\end{equation*}
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\end{itemize}
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\subsubsection*{Auswertungsstrategien}
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\begin{itemize}
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\item applikativ (\textit{leftmost-innermost}) \app \defin{3.13}{33}
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@ -77,16 +88,35 @@
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\item Church: Annotation der Variablen mit Typen, nur herleitbare Terme hinschreibbar
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\item Curry: Alle Terme hinschreibbar, dann Aussondern der nicht typisierbaren
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\item Typregeln: \siehe{S. 39}
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\begin{itemize}
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\item \textbf{TODO}
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\end{itemize}
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\begin{prooftree}
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\AxiomC{}
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\LeftLabel{$(Ax)$}
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\RightLabel{$x:\alpha \in \Gamma$}
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\UnaryInfC{$\Gamma \vdash x:\alpha$}
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\end{prooftree}
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\begin{prooftree}
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\AxiomC{$\Gamma \vdash t:\alpha\rightarrow \beta$}
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\AxiomC{$\Gamma \vdash s:\alpha$}
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\LeftLabel{$(\rightarrow_e)$}
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\BinaryInfC{$\Gamma \vdash ts:\beta$}
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\end{prooftree}
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\begin{prooftree}
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\AxiomC{$\Gamma, x:\alpha \vdash t:\beta$}
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\LeftLabel{$(\rightarrow_i)$}
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\UnaryInfC{$\Gamma \vdash \lambda x.t:\alpha\rightarrow\beta$}
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\end{prooftree}
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\item Typisierungsprobleme \siehe{S. 40}
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\begin{itemize}
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\item Typcheck: \enquote{Gilt $\Gamma \vdash t:\alpha$?}
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\item Typinferenz: \enquote{Was ist das beste $\alpha$ / Existiert $\alpha$ mit $\Gamma \vdash t:\alpha$?}
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\item Type inhabitation: \enquote{Existiert $t$ mit $\Gamma t:\alpha$?}
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\end{itemize}
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\item Inversionslemma \lem{3.29}{41} \textbf{TODO}
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\item Inversionslemma \lem{3.29}{41}
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\begin{enumerate}
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\item $\Gamma \vdash x:\alpha \Rightarrow x:\alpha\in\Gamma$
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\item $\Gamma \vdash ts:\beta \Rightarrow \exists\alpha$ mit $\Gamma \vdash t:\alpha\rightarrow\beta$ und $\Gamma\vdash s:\alpha$
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\item $\Gamma \vdash\lambda x.t:\gamma \Rightarrow \gamma = \alpha\rightarrow \beta$ mit $\Gamma, x:\alpha \vdash t:\beta$
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\end{enumerate}
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\item Typinferenz \siehe{S. 41}
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\begin{itemize}
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\item Typsubstitution $\sigma$ ist Lösung von $\Gamma \vdash t:\alpha$, wenn $\Gamma\sigma \vdash t:\alpha\sigma$ herleitbar
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@ -143,28 +173,79 @@
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\begin{align*}
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\alpha, \beta \coloneqq a\ \vert\ \alpha \rightarrow \beta\ \vert\ \forall a.\alpha && (a\in V)
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\end{align*}
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\item Typisierung: \textbf{TODO: $\lambda \rightarrow$}, \textbf{TODO} \defin{5.1}{84}
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\begin{align*}
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Inhalt...
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\end{align*}
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\item Typisierung: \defin{5.1}{84}
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\begin{prooftree}
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\AxiomC{}
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\LeftLabel{$(Ax)$}
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\RightLabel{$(x:\alpha \in \Gamma)$}
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\UnaryInfC{$\Gamma \vdash x:\alpha$}
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\end{prooftree}
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\begin{prooftree}
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\AxiomC{$\Gamma \vdash t:\alpha\rightarrow \beta$}
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\AxiomC{$\Gamma \vdash s:\alpha$}
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\LeftLabel{$(\rightarrow_e)$}
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\BinaryInfC{$\Gamma \vdash ts:\beta$}
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\end{prooftree}
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\begin{prooftree}
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\AxiomC{$\Gamma, x:\alpha \vdash t:\beta$}
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\LeftLabel{$(\rightarrow_i)$}
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\UnaryInfC{$\Gamma \vdash \lambda x.t:\alpha\rightarrow\beta$}
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\end{prooftree}
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\begin{prooftree}
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\AxiomC{$\Gamma \vdash s:\alpha$}
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\AxiomC{$a\notin FV(\Gamma)$}
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\LeftLabel{$(\forall_i)$}
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\BinaryInfC{$\Gamma \vdash s:\forall a.\alpha$}
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\end{prooftree}
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\begin{prooftree}
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\AxiomC{$\Gamma \vdash s:\forall a.\alpha$}
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\LeftLabel{$(\forall_e)$}
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\UnaryInfC{$\Gamma \vdash s:(\alpha[\beta/a])$}
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\end{prooftree}
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\end{itemize}
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\subsection*{Church-Kodierung}\siehe{S. 84}
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\begin{itemize}
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\item Natürliche Zahlen
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\begin{align*}
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\textbf{Todo}
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\mathbb{N}\ &\coloneqq \forall a.(a\rightarrow a) \rightarrow a \rightarrow a\\
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zero\ &: \mathbb{N}\\
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zero\ &= \lambda fx.x\\
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suc\ &: \mathbb{N}\rightarrow\mathbb{N}\\
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suc\ &= \lambda nfx.f(nfx)\\
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fold\ &: \forall a.(a\rightarrow a) \rightarrow a \rightarrow \mathbb{N}\rightarrow a\\
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fold\ &= \lambda fxn.nfx\\
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add &: \mathbb{N}\rightarrow\mathbb{N}\rightarrow\mathbb{N}\\
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add &= \lambda n.fold\ suc\ n
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\end{align*}
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\item Paare
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\begin{align*}
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Inhalt...
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(a\times b)\ &\coloneqq \forall r.(a\rightarrow b\rightarrow r)\rightarrow r\\
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pair\ &: \forall ab.a\rightarrow b\rightarrow (a\times b)\\
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pair\ &= \lambda xyf.fxy\\
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fst\ &: \forall ab.(a\times b) \rightarrow a\\
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fst\ &= \lambda p.p (\lambda xy.x)\\
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snd\ &: \forall ab.(a\times b) \rightarrow b\\
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snd\ &= \lambda p.p(\lambda xy.y)
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\end{align*}
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\item Summen
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\begin{align*}
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Inhalt...
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(a+b)\ &\coloneqq \forall r.(a\rightarrow r) \rightarrow (b \rightarrow r)\rightarrow r\\
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inl\ &: \forall ab.a\rightarrow(a+b)\\
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inl\ &= \lambda xfg.fx\\
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inr\ &: \forall ab.b\rightarrow(a+b)\\
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inr\ &= \lambda yfg.gy\\
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case\ &: \forall abs.(a\rightarrow s)\rightarrow (b\rightarrow s) \rightarrow (a+b) \rightarrow s\\
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case\ &= \lambda fgs.sfg
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\end{align*}
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\item Listen
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\begin{align*}
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Inhalt...
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List\ a\ &\coloneqq \forall r.r\rightarrow (a\rightarrow r\rightarrow r)\rightarrow r\\
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Nil\ &: \forall a.Lista\ a\\
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Nil\ &= \lambda uf.u\\
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Cons\ &: \forall a.a\rightarrow List\ a\rightarrow List\ a\\
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Cons\ &= \lambda xluf.fx(luf)\\
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len\ &: \forall a.List\ a\rightarrow \mathbb{N}\\
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len\ &= \lambda l.l\ zero\ (\lambda xr.suc\ r)
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\end{align*}
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\end{itemize}
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\subsection*{ML-Polymorphie}
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@ -187,11 +268,55 @@
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\Gamma &= ( x_1:S_1,\ldots, x_n :S_n )\\
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Cl(\Gamma, \alpha) &= \forall a_1,\ldots,a_k.\alpha &&\text{für $FV(\alpha)\backslash FV(\Gamma) = \lbrace a_1,\ldots,a_k \rbrace$}
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\end{align*}
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\item Typisierungsregeln \textbf{TODO}
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\item Inversionslemma \textbf{TODO}
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\item erweiterter Algorithmus W \textbf{TODO}
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\item Typisierungsregeln \siehe{S. 88}
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\begin{prooftree}
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\AxiomC{}
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\LeftLabel{$(\forall_e)$}
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\RightLabel{$(x:\forall a_1,\ldots, \forall a_k .\alpha) \in \Gamma$}
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\UnaryInfC{$\Gamma \vdash x:\alpha[\beta_1/a_1,\ldots, \beta_k/a_k]$}
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\end{prooftree}
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\begin{prooftree}
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\AxiomC{$\Gamma, x:\alpha \vdash t:\beta$}
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\LeftLabel{$(\rightarrow_i)$}
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\UnaryInfC{$\Gamma \vdash \lambda x.t:\alpha\rightarrow\beta$}
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\end{prooftree}
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\begin{prooftree}
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\AxiomC{$\Gamma \vdash t:\alpha\rightarrow \beta$}
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\AxiomC{$\Gamma \vdash s:\alpha$}
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\LeftLabel{$(\rightarrow_e)$}
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\BinaryInfC{$\Gamma \vdash ts:\beta$}
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\end{prooftree}
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\begin{prooftree}
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\AxiomC{$\Gamma \vdash t:\alpha$}
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\AxiomC{$\Gamma, x:Cl(\Gamma, \alpha)\vdash s:\beta$}
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\LeftLabel{(let)}
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\BinaryInfC{$\Gamma \vdash$ let $x = t$ in $s:\beta$}
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\end{prooftree}
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\item Inversionslemma \siehe{S. 89}
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\begin{enumerate}
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\item Wenn $\Gamma \vdash x:\alpha$, dann existieren Typen $\beta_i$ und ein Typschema $S = \forall a_1 \ldots \forall a_k.\gamma$, so dass $(x:S)\in \Gamma$ und $\alpha = \gamma[\beta_1/a_1,\ldots,\beta_k/a_k]$
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\item Wenn $\Gamma \vdash ($ let $x = s$ in $t) : \alpha$, dann existiert ein Typ $\beta$ nut $\Gamma \vdash s:\beta$ und $\Gamma, x:Cl(\Gamma, \beta) \vdash t:\alpha$
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\end{enumerate}
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\item erweiterter Algorithmus W mit Menge $PT$ von Typgleichungen
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\begin{align*}
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PT(\Gamma;x;\alpha) &= \lbrace a\pteq \gamma [a_1'/a_1,\ldots,a_k'/a_k] \vert (x: \forall a_1,\ldots,\forall a_k .\gamma) \in \Gamma\rbrace\\
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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}\\
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PT(\Gamma;ts;\alpha) &= PT(\Gamma; t; a\rightarrow \alpha ) \cup PT(\Gamma;s;a)\ \text{mit $a$ frisch}\\
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PT(\Gamma;(\text{let } x=s\text{ in }t);\alpha) &= PT(\Gamma\sigma, x:Cl(\Gamma\sigma, \sigma(b));t;\alpha\sigma)
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\end{align*}
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wobei $\sigma = mgu(PT(\Gamma;s;b))$ mit $b$ frisch
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\end{itemize}
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\section*{Reguläre Ausdrücke}
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\section*{Minimierung von deterministischen endlichen Automaten}
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\begin{enumerate}
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\item Entferne aus $Q$ alle nicht erreichbaren Zustände
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\item Initialisiere $R$ auf $\lbrace (q_1, q_2)\ \vert\ q_1 \in F \Leftrightarrow q_2 \in F\rbrace$
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\item Suche ein Paar $(q_1,q_2) \in R$ und einen Buchstaben $a \in \Sigma$ mit
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\begin{equation*}
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(\delta(a,q_1),\delta(a,q_2)) \notin R
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\end{equation*}
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Wenn kein solches Paar gefunden wird, gehe zu Schritt 4. Andernfalls entferne $(q_1,q_2)$ aus $R$ und fahre bei 3. fort.
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\item Identifiziere alle Zustandspaare in $R$.
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\end{enumerate}
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\section*{Unifikationsalgorithmus (Martelli/Montanari)}
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\begin{align*}
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&S \cup \lbrace x \pteq x\rbrace &&\rightarrow S &&\text{(delete)}\\
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