From 0174aa0cc6d2b90f1b9607c805acda0543fd271e Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Thu, 20 Mar 2014 13:42:52 +0100 Subject: [PATCH] Fixpunktkombinatoren; Haskell-Beispiele --- documents/Programmierparadigmen/Haskell.tex | 9 ++ documents/Programmierparadigmen/Lambda.tex | 109 +++++++++++++++++- .../Programmierparadigmen.pdf | Bin 724758 -> 742242 bytes .../scripts/haskell/Intersect.hs | 15 +++ .../scripts/haskell/fibonacci-stream.hs | 2 + 5 files changed, 129 insertions(+), 6 deletions(-) create mode 100644 documents/Programmierparadigmen/scripts/haskell/Intersect.hs create mode 100644 documents/Programmierparadigmen/scripts/haskell/fibonacci-stream.hs diff --git a/documents/Programmierparadigmen/Haskell.tex b/documents/Programmierparadigmen/Haskell.tex index 0d87b18..7b05d25 100644 --- a/documents/Programmierparadigmen/Haskell.tex +++ b/documents/Programmierparadigmen/Haskell.tex @@ -209,6 +209,11 @@ sich das ganze sogar noch kürzer schreiben: \inputminted[linenos, numbersep=5pt, tabsize=4, frame=lines, label=fibonacci-zip.hs]{haskell}{scripts/haskell/fibonacci-zip.hs} \inputminted[linenos, numbersep=5pt, tabsize=4, frame=lines, label=fibonacci-pattern-matching.hs]{haskell}{scripts/haskell/fibonacci-pattern-matching.hs} +Die unendliche Liste alle Fibonacci-Zahlen, also der Fibonacci-\textit{Stream}\xindex{Stream} +wird wie folgt erzeugt: + +\inputminted[linenos, numbersep=5pt, tabsize=4, frame=lines, label=fibonacci-stream.hs]{haskell}{scripts/haskell/fibonacci-stream.hs} + \subsection{Polynome}\xindex{Polynome}\xindex{zipWith}\xindex{foldr}\xindex{tail}% \inputminted[linenos, numbersep=5pt, tabsize=4, frame=lines, label=polynome.hs]{haskell}{scripts/haskell/polynome.hs} @@ -222,6 +227,10 @@ sich das ganze sogar noch kürzer schreiben: \inputminted[linenos, numbersep=5pt, tabsize=4, frame=lines, label=lauflaengencodierung.hs]{haskell}{scripts/haskell/lauflaengencodierung.hs} +\subsection{Intersections}\xindex{Intersections}% + +\inputminted[linenos, numbersep=5pt, tabsize=4, frame=lines, label=Intersect.hs]{haskell}{scripts/haskell/Intersect.hs} + \subsection{Funktionen höherer Ordnung}\xindex{Folds}\xindex{foldl}\xindex{foldr}\label{bsp:foldl-und-foldr} \inputminted[linenos, numbersep=5pt, tabsize=4, frame=lines, label=folds.hs]{haskell}{scripts/haskell/folds.hs} diff --git a/documents/Programmierparadigmen/Lambda.tex b/documents/Programmierparadigmen/Lambda.tex index 1ad6a5f..9eb10d2 100644 --- a/documents/Programmierparadigmen/Lambda.tex +++ b/documents/Programmierparadigmen/Lambda.tex @@ -46,12 +46,12 @@ Die Funktionsapplikation sei linksassoziativ. Es gilt also: Der untypisierte $\lambda$-Kalkül ist Turing-Äquivalent. \end{satz} -\section{Reduktionen}\index{Reduktionen|(} +\section{Reduktionen}\index{Reduktion|(} \begin{definition}[Redex]\xindex{Redex}% Eine $\lambda$-Term der Form $(\lambda x. t_1) t_2$ heißt Redex. \end{definition} -\begin{definition}[$\alpha$-Äquivalenz] +\begin{definition}[$\alpha$-Äquivalenz]\xindex{Reduktion!Alpha ($\alpha$)}\xindex{Äquivalenz!Alpha ($\alpha$)}% Zwei Terme $T_1, T_2$ heißen $\alpha$-Äquivalent, wenn $T_1$ durch konsistente Umbenennung in $T_2$ überführt werden kann. @@ -67,7 +67,7 @@ Die Funktionsapplikation sei linksassoziativ. Es gilt also: \end{align*} \end{beispiel} -\begin{definition}[$\beta$-Äquivalenz] +\begin{definition}[$\beta$-Äquivalenz]\xindex{Reduktion!Beta ($\beta$)}\xindex{Äquivalenz!Beta ($\beta$)}% Eine $\beta$-Reduktion ist die Funktionsanwendung auf einen Redex: \[(\lambda x. t_1) t_2 \Rightarrow t_1 [x \mapsto t_2]\] \end{definition} @@ -79,7 +79,7 @@ Die Funktionsapplikation sei linksassoziativ. Es gilt also: \end{defenum} \end{beispiel} -\begin{definition}[$\eta$-Äquivalenz] +\begin{definition}[$\eta$-Äquivalenz]\xindex{Reduktion!Eta ($\eta$)}\xindex{Äquivalenz!Eta ($\eta$)}% Zwei Terme $\lambda x. f~x$ und $f$ heißen $\eta$-Äquivalent, wenn $x$ nicht freie Variable von $f$ ist. \end{definition} @@ -87,7 +87,7 @@ Die Funktionsapplikation sei linksassoziativ. Es gilt also: \begin{beispiel}[$\eta$-Äquivalenz] TODO \end{beispiel} -\index{Reduktionen|)} +\index{Reduktion|)} \section{Auswertungsstrategien} \begin{definition}[Normalenreihenfolge]\xindex{Normalenreihenfolge}% @@ -193,4 +193,101 @@ Auch die gewohnten Operationen lassen sich so darstellen. \section{Weiteres} \begin{satz}[Satz von Curch-Rosser] Wenn zwei unterschiedliche Terme $a$ und $b$ äquivalent sind, d.h. mit Reduktionsschritten beliebiger Richtung ineinander transformiert werden können, dann gibt es einen weiteren Term $c$, zu dem sowohl $a$ als auch $b$ reduziert werden können. -\end{satz} \ No newline at end of file +\end{satz} + +\section{Fixpunktkombinator} +\begin{definition}[Fixpunkt]\xindex{Fixpunkt}% + Sei $f: X \rightarrow Y$ eine Funktion mit $\emptyset \neq A = X \cap Y$ und + $a \in A$. + + $a$ heißt \textbf{Fixpunkt} der Funktion $f$, wenn $f(a) = a$ gilt. +\end{definition} + +\begin{beispiel}[Fixpunkt] + \begin{bspenum} + \item $f_1: \mdr \rightarrow \mdr; f(x) = x^2 \Rightarrow x_1 = 0$ ist + Fixpunkt von $f$, da $f(0) = 0$. $x_2 = 1$ ist der einzige weitere + Fixpunkt dieser Funktion. + \item $f_2: \mdn \rightarrow \mdn$ hat ganz $\mdn$ als Fixpunkte, also + insbesondere unendlich viele Fixpunkte. + \item $f_3: \mdr \rightarrow \mdr; f(x) = x+1$ hat keinen einzigen Fixpunkt. + \item $f_4: \mdr[X] \rightarrow \mdr[X]; f(p) = p^2$ hat $p_1(x) = 0$ und + $p_2(x)=1$ als Fixpunkte. + \end{bspenum} +\end{beispiel} + +\begin{definition}[Kombinator]\xindex{Kombinator}% + Ein Kombinator ist eine Abbildung ohne freie Variablen. +\end{definition} + +\begin{beispiel}[Kombinatoren\footnotemark]% + \begin{bspenum} + \item $\lambda a.\ a$ + \item $\lambda a.\ \lambda b.\ a$ + \item $\lambda f.\ \lambda a.\ \lambda b. f\ b\ a$ + \end{bspenum} +\end{beispiel} +\footnotetext{Quelle: \url{http://www.haskell.org/haskellwiki/Combinator}} + +\begin{definition}[Fixpunkt-Kombinator]\xindex{Fixpunkt-Kombinator}% + Sei $f$ ein Kombinator, der $f\ g = g\ (f\ g)$ erfüllt. Dann heißt $f$ + \textbf{Fixpunktkombinator}. +\end{definition} + +Insbesondere ist also $f \ g$ ein Fixpunkt von $g$. + +\begin{definition}[Y-Kombinator]\xindex{Y-Kombinator}% + Der Fixpunktkombinator + \[Y := \lambda f.\ (\lambda x.\ f\ (x\ x))\ (\lambda x.\ f\ (x\ x))\] + heißt $Y$-Kombinator. +\end{definition} + +\begin{behauptung} + Der $Y$-Kombinator ist ein Fixpunktkombinator. +\end{behauptung} + +\begin{beweis}\footnote{Quelle: Vorlesung WS 2013/2014, Folie 175}\leavevmode + + \textbf{Teil 1:} Offensichtlich ist $Y$ ein Kombinator. + + \textbf{Teil 2:} z.~Z.: $Y f \Rightarrow^* f \ (Y \ f)$ + + \begin{align*} + Y\ f &=\hphantom{^\beta f\ } (\lambda f.\ (\lambda x.\ f\ (x\ x))\ (\lambda x.\ f\ (x\ x)))\ f\\ + &\Rightarrow^\beta\hphantom{f \ (\lambda f.\ } (\lambda x. f\ (x\ x))\ (\lambda x.\ f\ (x\ x))\\ + &\Rightarrow^\beta f \ (\hphantom{\lambda f.\ }(\lambda x.\ f\ (x\ x))\ (\lambda x.\ f\ (x\ x)))\\ + &\Rightarrow^\beta f \ (\lambda f.\ (\lambda x.\ f\ (x\ x))\ (\lambda x.\ f\ (x\ x))\ f)\\ + &=\hphantom{^\beta} f \ (Y \ f) + \end{align*} + $\qed$ +\end{beweis} + +\begin{definition}[Turingkombinator]\xindex{Turingkombinator}% + Der Fixpunktkombinator + \[\Theta := (\lambda x. \lambda y. y\ (x\ x\ y)) (\lambda x.\ \lambda y.\ y\ (x\ x\ y))\] + heißt \textbf{Turingkombinator}. +\end{definition} + +\begin{behauptung} + Der Turing-Kombinator $\Theta$ ist ein Fixpunktkombinator. +\end{behauptung} + +\begin{beweis}\footnote{Quelle: Übungsblatt 6, WS 2013/2014} + + \textbf{Teil 1:} Offensichtlich ist $\Theta$ ein Kombinator. + + \textbf{Teil 2:} z.~Z.: $\Theta f \Rightarrow^* f \ (\Theta \ f)$ + + Sei $\Theta_0 := (\lambda x.\ \lambda y.\ y\ (x\ x\ y))$. Dann gilt: + \begin{align*} + \Theta\ f &= ((\lambda x.\ \lambda y.\ y\ (x\ x\ y))\ \Theta_0)\ f\\ + &\Rightarrow^\beta (\lambda y. y\ (\Theta_0 \ \Theta_0 \ y))\ f\\ + &\Rightarrow^\beta f \ (\Theta_0 \Theta_0 f)\\ + &= f \ (\Theta \ f) + \end{align*} + $\qed$ +\end{beweis} + + + +Für den Turing-Operator gilt \ No newline at end of file diff --git a/documents/Programmierparadigmen/Programmierparadigmen.pdf b/documents/Programmierparadigmen/Programmierparadigmen.pdf index fa7f2b12d7dfc6e77aae33a2a112e60f1e12304a..6e709b654145ce3d6bd72285b6379c0e22c67e4c 100644 GIT binary patch delta 201656 zcmZU)V~l3a7d<*{PTRJoZQHhO+j`ozZQHhO+qN;?edqoCFFxF)PO?s=a#Bg9&f0se zy;E|i4@zj_M1Yu62_bMEI?p+Jru!nfhp<6#TNwQ6^am7_v)v^@H_df!>tX#9JLw%Sf?@6RQGSE zSq4%!jn?jGjfQR3y?Z0&YJcMu$crl~cBT*2j$Dzo}$)mgt zIcC<)E0(+qh3K845h2s>MHwLhgN9G_&3PXfu=4^*AW`pgzz)X{NMf-fNO?}i=*J-< z#V6082@*9hLXHTaZh!^=g76Ic9z~=S^zzo(xMd+^v88HyZ_yg(tX%Pa0 zMA!oGwPb-ff}X7j8V)rHpe#=@!j5rb2qW$Njq19PM&m^3{R95o&Cr1!A@cU6KY~p` 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a/documents/Programmierparadigmen/scripts/haskell/fibonacci-stream.hs b/documents/Programmierparadigmen/scripts/haskell/fibonacci-stream.hs new file mode 100644 index 0000000..baa3f81 --- /dev/null +++ b/documents/Programmierparadigmen/scripts/haskell/fibonacci-stream.hs @@ -0,0 +1,2 @@ +fibs :: [Integer] +fibs = 0 : 1 : zipWith (+) fibs (tail fibs) \ No newline at end of file