From 2f943ff66785629494334ca22a3bb7c816eebedb Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Wed, 12 Mar 2014 21:07:00 +0100 Subject: [PATCH] some haskell / chruch numbers --- documents/Programmierparadigmen/Haskell.tex | 4 ++ .../Programmierparadigmen.pdf | Bin 679508 -> 699908 bytes .../Programmiersprachen.tex | 49 ++++++++++++++++-- .../scripts/haskell/church.hs | 8 +++ 4 files changed, 58 insertions(+), 3 deletions(-) create mode 100644 documents/Programmierparadigmen/scripts/haskell/church.hs diff --git a/documents/Programmierparadigmen/Haskell.tex b/documents/Programmierparadigmen/Haskell.tex index 1600da2..33d9c44 100644 --- a/documents/Programmierparadigmen/Haskell.tex +++ b/documents/Programmierparadigmen/Haskell.tex @@ -209,9 +209,13 @@ sich das ganze sogar noch kürzer schreiben: \inputminted[linenos, numbersep=5pt, tabsize=4, frame=lines, label=fibonacci-pattern-matching.hs]{haskell}{scripts/haskell/fibonacci-pattern-matching.hs} \subsection{Quicksort} +TODO + \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} +\subsection{Chruch-Zahlen} +\inputminted[linenos, numbersep=5pt, tabsize=4, frame=lines, label=church.hs]{haskell}{scripts/haskell/church.hs} \subsection{Standard Prelude} Hier sind die Definitionen eininger wichtiger Funktionen: diff --git a/documents/Programmierparadigmen/Programmierparadigmen.pdf b/documents/Programmierparadigmen/Programmierparadigmen.pdf index fb37f77110ce15eea5a148bcc1bb96d93c9b3631..007617ca854164b615d9843c7babdac0e29b1eb3 100644 GIT binary patch delta 231220 zcmZsBV|1_0w`6SFwr$(CZRf=J#XhlZ+qP}z#I|*k%=`ZDojYsR%(q&*pVj?kcRf|r zd%2rBWSb^l9E61_Ng0+3kfN~_yTyUz_o_K<`Vg;wuK+9@V3)*Zg+f*n3_EkZpgnI4l& zz7I|4gBIHi4B?oYJfN#EDa(@BtYn5}eTPE_*oyJoI36ZVsHqNc zl+`f(lORq;AedX-4@w)5JxoPfHg4`Y1c}_gPHYvI@Jb7I+LV3xs*}KZBz)UYfXB*x 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Man erhält + + \begin{align*} + \sigma(A_1) &= \left(a, b, f(b) \right)\\ + \sigma(A_2) &= \left(a, b, f(b) \right) + \end{align*} + + mit + \[\sigma = \{X \mapsto a, Y \mapsto b, Z \mapsto f(b)\}\] \end{beispiel} + +\begin{definition}[Allgemeinster Unifikator]\xindex{Unifikator!allgemeinster}% + Ein Unifikator $\sigma$ heißt \textit{allgemeinster Unifikator}, wenn + es für jeden Unifikator $\gamma$ eine Substitution $\delta$ mit + \[\gamma = \delta \circ \sigma\] + gibt. +\end{definition} + +\begin{beispiel}[Allgemeinster Unifikator\footnotemark] + Sei + \[C = \Set{f(a,D) = Y, X = g(b), g(Z) = X}\] + eine Menge von Gleichungen über Terme. + + Dann ist + \[\gamma = [Y \text{\pointer} f(a,b), D \text{\pointer} b, X \text{\pointer} g(b), Z \text{\pointer} b]\] + ein Unifikator für $C$. Jedoch ist + \[\sigma = [Y \text{\pointer} f(a,D), X \text{\pointer} g(b), Z \text{\pointer} b]\] + der allgemeinste Unifikator. Mit + \[\delta = [D \text{\pointer} b]\] + gilt $\gamma = \delta \circ \sigma$. +\end{beispiel} +\footnotetext{Folie 268 von Prof. Snelting} + +\footnotetext{\url{https://de.wikipedia.org/w/index.php?title=Unifikation\_(Logik)&oldid=116848554\#Beispiel}} \ No newline at end of file diff --git a/documents/Programmierparadigmen/scripts/haskell/church.hs b/documents/Programmierparadigmen/scripts/haskell/church.hs new file mode 100644 index 0000000..5b551eb --- /dev/null +++ b/documents/Programmierparadigmen/scripts/haskell/church.hs @@ -0,0 +1,8 @@ +type Church t = (t -> t) -> t -> t + +int2church :: Integer -> Church t +int2church 0 s z = z +int2church n s z = int2church (n - 1) s (s z) + +church2int :: Church Integer -> Integer +church2int n = n (+1) 0 \ No newline at end of file