From 2306a2af2e95ca3e7d7fb0314337341ddf5bd004 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Mon, 24 Feb 2014 11:44:57 +0100 Subject: [PATCH] Haskell / Prolog --- .../Programmierparadigmen/Definitionen.tex | 118 ++++++++++++++++++ documents/Programmierparadigmen/Haskell.tex | 10 +- .../Programmierparadigmen.pdf | Bin 483576 -> 505389 bytes .../Programmierparadigmen.tex | 2 + .../Programmiersprachen.tex | 15 ++- documents/Programmierparadigmen/Prolog.tex | 8 ++ .../programmierparadigmen.sublime-project | 8 ++ .../haskell/binomialkoeffizient-guard.hs | 5 + .../scripts/prolog/hello-world.pl | 2 + documents/Programmierparadigmen/shortcuts.sty | 22 +--- 10 files changed, 166 insertions(+), 24 deletions(-) create mode 100644 documents/Programmierparadigmen/Definitionen.tex create mode 100644 documents/Programmierparadigmen/programmierparadigmen.sublime-project create mode 100644 documents/Programmierparadigmen/scripts/haskell/binomialkoeffizient-guard.hs create mode 100644 documents/Programmierparadigmen/scripts/prolog/hello-world.pl diff --git a/documents/Programmierparadigmen/Definitionen.tex b/documents/Programmierparadigmen/Definitionen.tex new file mode 100644 index 0000000..c54bcd8 --- /dev/null +++ b/documents/Programmierparadigmen/Definitionen.tex @@ -0,0 +1,118 @@ +%!TEX root = Programmierparadigmen.tex +\markboth{Ergänzende Definitionen}{Ergänzende Definitionen} +\chapter*{Ergänzende Definitionen} +\addcontentsline{toc}{chapter}{Ergänzende Definitionen} + +\begin{definition}[Quantoren]\xindex{Quantor}% + \begin{defenum} + \item $\forall x \in X: p(x)$: Für alle Elemente $x$ aus der Menge $X$ gilt + die Aussage $p$. + \item $\exists x \in X: p(x)$: Es gibt mindestens ein Element $x$ aus der + Menge $X$, für das die Aussage $p$ gilt. + \item $\exists! x \in X: p(x)$: Es gibt genau ein Element $x$ in der + Menge $X$, sodass die Aussage $p$ gilt. + \end{defenum} +\end{definition} + +\begin{definition}[Prädikatenlogik]\xindex{Prädikatenlogik}% + Eine Prädikatenlogik ist ein formales System, das Variablen und Quantoren + nutzt um Aussagen zu formulieren. +\end{definition} + +\begin{definition}[Aussagenlogik]\xindex{Aussagenlogik}% + TODO +\end{definition} + +\begin{definition}[Grammatik]\xindex{Grammatik}\xindex{Alphabet}\xindex{Nichtterminal}\xindex{Startsymbol}\xindex{Produktionsregel}\index{Ableitungsregel|see{Produktionsregel}}% + Eine (formale) \textbf{Grammatik} ist ein Tupel $(\Sigma, V, P, S)$ wobei gilt: + \begin{defenumprops} + \item $\Sigma$ ist eine endliche Menge und heißt \textbf{Alphabet}, + \item $V$ ist eine endliche Menge mit $V \cap \Sigma = \emptyset$ und + heißt \textbf{Menge der Nichtterminale}, + \item $S \in V$ heißt das \textbf{Startsymbol} + \item $P = \Set{p: I \rightarrow r | I \in (V \cup \Sigma)^+, r \in (V \cup \Sigma)^*}$ ist eine endliche Menge aus \textbf{Produktionsregeln} + \end{defenumprops} +\end{definition} + +Man schreibt: + +\begin{itemize} + \item $a \Rightarrow b$: Die Anwendung einer Produktionsregel auf $a$ ergibt $b$. + \item $a \Rightarrow^* b$: Die Anwendung mehrerer (oder keiner) Produktionsregeln auf + $a$ ergibt $b$. + \item $a \Rightarrow^+ b$: Die Anwendung mindestens einer Produktionsregel auf $a$ + ergibt $b$. +\end{itemize} + +\begin{beispiel}[Formale Grammatik] + Folgende Grammatik $G = (\Sigma, V, P, A)$ erzeugt alle korrekten Klammerausdrücke: + + \begin{itemize} + \item $\Sigma = \Set{(, )}$ + \item $V = \Set{\alpha}$ + \item $s = \alpha$ + \item $P = \Set{\alpha \rightarrow () | \alpha \alpha | (\alpha)}$ + \end{itemize} +\end{beispiel} + +\begin{definition}[Kontextfreie Grammatik]\xindex{Grammatik!Kontextfreie}% + Eine Grammatik $(\Sigma, V, P, S)$ heißt \textbf{kontextfrei}, wenn für + jede Produktion $p: I \rightarrow r$ gilt: $I \in V$. +\end{definition} + +\begin{definition}[Sprache]\xindex{Sprache}% + Sei $G = (\Sigma, V, P, S)$ eine Grammatik. Dann ist + \[L(G) := \Set{\omega \in \Sigma^* | S \Rightarrow^* \omega}\] + die Menge aller in der Grammatik ableitbaren Wörtern. $L(G)$ heißt Sprache + der Grammatik $G$. +\end{definition} + +\begin{definition}\xindex{Linksableitung}\xindex{Rechtsableitung}% + Sei $G = (\Sigma, V, P, S)$ eine Grammatik und $a \in (V \cup \Sigma)^+$. + \begin{defenum} + \item $\Rightarrow_L$ heißt \textbf{Linksableitung}, wenn die Produktion + auf das linkeste Nichtterminal angewendet wird. + \item $\Rightarrow_R$ heißt \textbf{Rechtsableitung}, wenn die Produktion + auf das rechteste Nichtterminal angewendet wird. + \end{defenum} +\end{definition} + +\begin{beispiel}[Links- und Rechtsableitung] + Sie $G$ wie zuvor die Grammatik der korrekten Klammerausdrücke: + + \begin{equation*} + \begin{aligned}[t] + \alpha &\Rightarrow_L \alpha \alpha\\ + &\Rightarrow_L \alpha \alpha \alpha\\ + &\Rightarrow_L () \alpha \alpha\\ + &\Rightarrow_L () (\alpha) \alpha\\ + &\Rightarrow_L () (()) \alpha\\ + &\Rightarrow_L () (()) ()\\ + \end{aligned} + \qquad\Longleftrightarrow\qquad + \begin{aligned}[t] + \alpha &\Rightarrow_R \alpha \alpha\\ + &\Rightarrow_R \alpha \alpha \alpha\\ + &\Rightarrow_R \alpha \alpha ()\\ + &\Rightarrow_R \alpha (\alpha) ()\\ + &\Rightarrow_R \alpha (()) ()\\ + &\Rightarrow_R () (()) ()\\ + \end{aligned} + \end{equation*} +\end{beispiel} + +\begin{definition}[LL($k$)-Grammatik]\xindex{LL(k)-Grammatik}% + Sei $G = (\Sigma, V, P, S)$ eine kontextfreie Grammatik. $G$ heißt + LL($k$)-Grammatik für $k \in \mathbb{N}_{\geq 1}$, wenn jeder Ableitungsschritt durch + die linkesten $k$ Symbole der Eingabe bestimmt ist.\todo{Was ist die Eingabe einer Grammatik?} +\end{definition} + +Ein LL-Parser ist ein Top-Down-Parser liest die Eingabe von Links nach rechts +und versucht eine Linksableitung der Eingabe zu berechnen. Ein LL($k$)-Parser +kann $k$ Token vorausschauen, wobei $k$ als \textit{Lookahead}\xindex{Lookahead} +bezeichnet wird. + +\begin{satz} + Für linksrekursive, kontextfreie Grammatiken $G$ gilt: + \[\forall k \in \mathbb{N}: G \notin \SLL(k)\] +\end{satz} \ No newline at end of file diff --git a/documents/Programmierparadigmen/Haskell.tex b/documents/Programmierparadigmen/Haskell.tex index c1075e8..d369b66 100644 --- a/documents/Programmierparadigmen/Haskell.tex +++ b/documents/Programmierparadigmen/Haskell.tex @@ -28,6 +28,8 @@ Speichere folgenden Quelltext als \texttt{hello-world.hs}: Kompiliere ihn mit \texttt{ghc -o hello hello-world.hs}. Es wird eine ausführbare Datei erzeugt. +Alternativ kann es direkt mit \texttt{runghc hello-world.hs} ausgeführt werden. + \section{Syntax} \subsection{Klammern und Funktionsdeklaration} Haskell verzichtet an vielen Stellen auf Klammern. So werden im @@ -57,15 +59,16 @@ Zu lesen ist die Deklaration wie folgt: \end{itemize} \end{itemize} -\todo[inline]{Gibt es Funktionsdeklarationen, die äquivalent? (bis auf wechsel des namens und der Reihenfolge)} +\todo[inline]{Gibt es Funktionsdeklarationen, die bis auf Wechsel des Namens und der Reihenfolge äquivalent sind?} \subsection{if / else} Das folgende Beispiel definiert den Binomialkoeffizienten (vgl. \cref{bsp:binomialkoeffizient}) \inputminted[numbersep=5pt, tabsize=4]{haskell}{scripts/haskell/binomialkoeffizient.hs} -\inputminted[numbersep=5pt, tabsize=4]{bash}{scripts/haskell/compile-binom.sh} -\todo[inline]{Guards} +Das könnte man auch mit sog. Guards machen:\xindex{Guard} + +\inputminted[numbersep=5pt, tabsize=4]{haskell}{scripts/haskell/binomialkoeffizient-guard.hs} \subsection{Rekursion} Die Fakultätsfunktion wurde wie folgt implementiert: @@ -86,6 +89,7 @@ hat einen Speicherverbrauch von $\mathcal{O}(n)$. Durch einen \item \texttt{[1,2,3]} erzeugt eine Liste mit den Elementen $1, 2, 3$ \item \texttt{:} wird \textbf{cons}\xindex{cons} genannt und ist der Listenkonstruktor. + \item \texttt{list !! i} gibt das $i$-te Element von \texttt{list} zurück. \item \texttt{head list} gibt den Kopf von \texttt{list} zurück, \texttt{tail list} den Rest: \inputminted[numbersep=5pt, tabsize=4]{haskell}{scripts/haskell/list-basic.sh} diff --git a/documents/Programmierparadigmen/Programmierparadigmen.pdf b/documents/Programmierparadigmen/Programmierparadigmen.pdf index 147a9567c54884723ce80a1d26b8ed4a7b1631d8..344948cf972563bff3fff96ed6ed22a6e2c015ee 100644 GIT binary patch delta 221158 zcmZs?Q*b6s*M=M0*2Kxgwr$(?#P$>0Pi)(^ZQHh!iD$q6cewYi>Vw8fSFf(F>t5^b zw}D?Z*MtdT;7nYczy}N}K#GRE!?pxM&t2UGHt9@#q=YpR-Bq85uMuX8H7$}lggriesDEXmipx*4C7 z8A~7+MWaxAa#q#TB`NGc+?C{{O;g<)bQK$P^CQ_Ew_dT#fK zH*vfiscmAn+A*_8GVyV6kx_ksAR)Xb7#7+H99ffyb7IaZ5h{<#G5Cm7MG%Ry37-gT 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index b19ac15..3f8a803 100644 --- a/documents/Programmierparadigmen/Programmierparadigmen.tex +++ b/documents/Programmierparadigmen/Programmierparadigmen.tex @@ -105,6 +105,8 @@ \clearpage \input{Abkuerzungen} \clearpage +\input{Definitionen} +\clearpage \input{Symbolverzeichnis} \clearpage \addcontentsline{toc}{chapter}{Stichwortverzeichnis} diff --git a/documents/Programmierparadigmen/Programmiersprachen.tex b/documents/Programmierparadigmen/Programmiersprachen.tex index 1dd7618..79e46b8 100644 --- a/documents/Programmierparadigmen/Programmiersprachen.tex +++ b/documents/Programmierparadigmen/Programmiersprachen.tex @@ -1,3 +1,4 @@ +%!TEX root = Programmierparadigmen.tex \chapter{Programmiersprachen} Im folgenden werden einige Begriffe definiert anhand derer Programmiersprachen unterschieden werden können. @@ -73,7 +74,7 @@ Wichtige Vorteile von funktionalen Programmiersprachen sind: \end{itemize} \begin{definition}[Logisches Paradigma]\xindex{Programmierung!logische} - In der logischen Programmierung baut man Unifikation.\todo{genauer!} + In der logischen Programmierung baut auf der Unifikation auf.\todo{genauer!} \end{definition} \section{Typisierung} @@ -99,12 +100,18 @@ obwohl es Programmiersprachen gibt, die beides unterstützen. C und Java werden kompiliert, Python und TCL interpretiert. \section{Dies und das} -\begin{definition}[Seiteneffekt]\xindex{Seiteneffekt}\xindex{Nebeneffekt}\xindex{Wirkung} +\begin{definition}[Seiteneffekt]\xindex{Seiteneffekt}\index{Nebeneffekt|see{Seiteneffekt}}\index{Wirkung|see{Seiteneffekt}}% Seiteneffekte sind Veränderungen des Zustandes.\todo{Das geht besser} \end{definition} Manchmal werden Seiteneffekte auch als Nebeneffekt oder Wirkung bezeichnet. -\begin{definition}[Unifikation]\xindex{Unifikation} - \todo[inline]{Was ist das?} +\begin{definition}[Unifikation]\xindex{Unifikation}% + Die Unifikation ist eine Operation in der Logik und dient zur Vereinfachung + prädikatenlogischer Ausdrücke. + \todo[inline]{Das ist keine formale Definition!} \end{definition} + +\begin{beispiel}[Unifikation] + +\end{beispiel} diff --git a/documents/Programmierparadigmen/Prolog.tex b/documents/Programmierparadigmen/Prolog.tex index 35eeeee..703930c 100644 --- a/documents/Programmierparadigmen/Prolog.tex +++ b/documents/Programmierparadigmen/Prolog.tex @@ -7,6 +7,14 @@ befolgt. Eine interaktive Prolog-Sitzung startet man mit \texttt{swipl}. In Prolog definiert man Terme. +\section{Erste Schritte} +\subsection{Hello World} +Speichere folgenden Quelltext als \texttt{hello-world.pl}: +\inputminted[linenos, numbersep=5pt, tabsize=4, frame=lines, label=hello-world.hs]{prolog}{scripts/prolog/hello-world.hs} + +Kompiliere ihn mit \texttt{gplc hello-world.pl}. Es wird eine +ausführbare Datei erzeugt. + \section{Syntax} \section{Beispiele} \subsection{Humans} diff --git a/documents/Programmierparadigmen/programmierparadigmen.sublime-project b/documents/Programmierparadigmen/programmierparadigmen.sublime-project new file mode 100644 index 0000000..4676b2a --- /dev/null +++ b/documents/Programmierparadigmen/programmierparadigmen.sublime-project @@ -0,0 +1,8 @@ +{ + "folders": + [ + { + "path": "/home/moose/Downloads/LaTeX-examples/documents/Programmierparadigmen" + } + ] +} diff --git a/documents/Programmierparadigmen/scripts/haskell/binomialkoeffizient-guard.hs b/documents/Programmierparadigmen/scripts/haskell/binomialkoeffizient-guard.hs new file mode 100644 index 0000000..870050d --- /dev/null +++ b/documents/Programmierparadigmen/scripts/haskell/binomialkoeffizient-guard.hs @@ -0,0 +1,5 @@ +binom :: (Eq a, Num a, Num a1) => a -> a -> a1 +binom n k + | (k==0) || (k==n) = 1 + | otherwise = binom (n-1) (k-1) + + binom (n-1) k \ No newline at end of file diff --git a/documents/Programmierparadigmen/scripts/prolog/hello-world.pl b/documents/Programmierparadigmen/scripts/prolog/hello-world.pl new file mode 100644 index 0000000..b070ba0 --- /dev/null +++ b/documents/Programmierparadigmen/scripts/prolog/hello-world.pl @@ -0,0 +1,2 @@ +:- initialization(main). +main :- write('Hello World!'), nl, halt. diff --git a/documents/Programmierparadigmen/shortcuts.sty b/documents/Programmierparadigmen/shortcuts.sty index 94c7c50..672256e 100644 --- a/documents/Programmierparadigmen/shortcuts.sty +++ b/documents/Programmierparadigmen/shortcuts.sty @@ -64,28 +64,12 @@ \def\mdn{\ensuremath{\mathbb{N}}} \def\mdh{\ensuremath{\mathbb{H}}} \def\gdw{\ensuremath{\Leftrightarrow}} - -\def\atlas{\ensuremath{\mathcal{A}}} -\DeclareMathOperator{\rang}{rg} - -\def\GL{\ensuremath{\mathrm{GL}}} -\def\SL{\ensuremath{\mathrm{SL}}} -\def\PSL{\ensuremath{\mathrm{PSL}}} \newcommand\mapsfrom{\mathrel{\reflectbox{\ensuremath{\mapsto}}}} \newcommand\dcup{\mathbin{\dot{\cup}}} \newcommand{\id}{\textnormal{id}} -\DeclareMathOperator{\Deck}{Deck} -\DeclareMathOperator{\Fix}{Fix} -\DeclareMathOperator{\Iso}{Iso} -\DeclareMathOperator{\grad}{grad} -\DeclareMathOperator{\Perm}{Perm} -\DeclareMathOperator{\Sym}{Sym} -\DeclareMathOperator{\Homoo}{\textnormal{Homöo}} -\DeclareMathOperator{\Diffeo}{Diffeo} \DeclareMathOperator{\conv}{conv} \DeclareMathOperator{\IWS}{IWS} -\DeclareMathOperator{\DV}{DV} -\DeclareMathOperator{\Rg}{Rg} +\DeclareMathOperator{\SLL}{SLL} \DeclareMathOperator{\Bild}{Bild} \newcommand{\iu}{{i\mkern1mu}} % imaginary unit %\DeclareMathOperator{\Re}{Re} @@ -116,6 +100,10 @@ \setlist[defenum]{label=\alph*),ref=\textup{\thedefinition.\alph*}} \crefalias{defenumi}{definition} +\newlist{defenumprops}{enumerate}{1} +\setlist[defenumprops]{label=(\roman*),ref=\textup{\thedefinition.\roman*}} +\crefalias{defenumpropsi}{definition} + \newlist{bemenum}{enumerate}{1} \setlist[bemenum]{label=\alph*),ref=\textup{\thebemerkung.\alph*}} \crefalias{bemenumi}{bemerkung}