From 8b909ca53fa2cc5531b5f1a9c7f1ef51b0cedd99 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Sun, 2 Mar 2014 19:35:37 +0100 Subject: [PATCH] =?UTF-8?q?Abschnitt=20'Typinferenz'=20hinzugef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/Programmierparadigmen/C.tex | 2 +- .../Programmierparadigmen.pdf | Bin 564385 -> 580452 bytes .../Programmierparadigmen.tex | 2 + .../Programmiertechniken.tex | 2 +- .../Programmierparadigmen/Typinferenz.tex | 98 ++++++++++++++++++ documents/Programmierparadigmen/Vorwort.tex | 9 +- .../scripts/python/multiplikation.py | 6 ++ 7 files changed, 115 insertions(+), 4 deletions(-) create mode 100644 documents/Programmierparadigmen/Typinferenz.tex create mode 100644 documents/Programmierparadigmen/scripts/python/multiplikation.py diff --git a/documents/Programmierparadigmen/C.tex b/documents/Programmierparadigmen/C.tex index 91031d1..561a3ca 100644 --- a/documents/Programmierparadigmen/C.tex +++ b/documents/Programmierparadigmen/C.tex @@ -29,7 +29,7 @@ zusätzlich kann man \texttt{char}\xindex{char} und \texttt{int}\xindex{int} noch in \texttt{signed}\xindex{signed} und \texttt{unsigned}\xindex{unsigned} unterscheiden. -\section{ASCII-Tabelle} +\section{ASCII-Tabelle}\label{sec:ascii-tabelle} \begin{table}[htp] \centering \begin{tabular}{|l|l||l|l||l|l||l|l|} diff --git a/documents/Programmierparadigmen/Programmierparadigmen.pdf b/documents/Programmierparadigmen/Programmierparadigmen.pdf index ab6dc21b89a689cdb3aa929d87d2d4f4bdc5d306..921b6c28d5040ab8b878f9199c583eff294c2922 100644 GIT binary patch delta 342391 zcmZttV{k4^&^3(4wr$(CZQHhe#r73r$F}Vq+qRwT?Cdx>_w#KkUq@6rj3b^P<3rN>ebl#Ri>ATjRVHMIjn+rmN0i#gd!B9$RWy33+h{#x5 z3&pTZ*3ds+e@S47WwJ{_BAFc7^>wz|yFr@8wfr5)5yYq~PC#?OO`XnaOI5~sSmGeH zl>!&rN}Yag5xlTDLt}P70TV%O9x|j(;`o>bL{i9@q*2rYf;Nn{1Eyu#1C`9G9L_Mz z_}!qce-W_3zM@ai1{^{rTkGKZDhiFD3r&E`5y?X9kj^9O~(S+OiM~zTND+dM6Ww*N$jY`%SW>0LIoC_#!d@4d!$fkgDq|p0}w~}vU2;S6XW4_k+zJ6 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So kann ein +\texttt{bool} durch ein einzelnes Bit repräsentiert werden oder eine Bitfolge +zugrunde liegen. + +Auf Typen sind Operationen definiert. So kann man auf numerischen Typen eine +Addition (+), eine Subtraktion (-), eine Multiplikation (*) und eine Division (/) +definieren.\\ +Ich schreibe hier bewusst \enquote{eine} Multiplikation und nicht \enquote{die} +Multiplikation, da es verschiedene Möglichkeiten gibt auf Gleitpunktzahlen +Multiplikationen zu definieren. So kann man beispielsweise die Assoziativität +unterschiedlich wählen. + +\begin{beispiel}[Multiplikation ist nicht assoziativ] + In Python 3 ist die Multiplikation linksassoziativ. Also: + \inputminted[numbersep=5pt, tabsize=4]{python}{scripts/python/multiplikation.py} +\end{beispiel} + +\begin{definition}[Typvariable]\xindex{Typvariable}% + Eine Typvariable repräsentiert einen Typen. +\end{definition} + +\underline{Hinweis:} Üblicherweise werden kleine griechische Buchstaben ($\alpha, \beta, \tau_1, \tau_2, \dots$) als Typvariablen gewählt. + +Genau wie Typen bestimmte Operationen haben, die auf ihnen definiert sind, +kann man sagen, dass Operationen bestimmte Typen, auf die diese Anwendbar sind. So ist +\[\alpha+\beta\] +für numerische $\alpha$ und $\beta$ wohldefiniert, auch wenn $\alpha$ und $\beta$ boolesch sind +oder beides Strings sind könnte das Sinn machen. Es macht jedoch z.~B. keinen Sinn, +wenn $\alpha$ ein String ist und $\beta$ boolesch. + +Die Menge aller Operationen, die auf die Variablen angewendet werden, nennt man +\textbf{Typkontext}\xindex{Typkontext}. Dieser wird üblicherweise mit $\Gamma$ +bezeichnet. + +Das Ableiten einer Typisierung für einen Ausdruck nennt man \textbf{Typinferenz}\xindex{Typinferenz}. +Man schreibt: $\vdash(\lambda x.2): \alpha \rightarrow \text{int}$. + +Bei solchen Ableitungen sind häufig viele Typen möglich. So kann der Ausdruck +\[\lambda x.2\] +Mit folgenderweise typisiert werden: +\begin{itemize} + \item $\vdash(\lambda x.2): \text{bool} \rightarrow int$ + \item $\vdash(\lambda x.2): \text{int} \rightarrow int$ + \item $\vdash(\lambda x.2): \text{Char} \rightarrow int$ + \item $\vdash(\lambda x.2): \alpha \rightarrow int$ +\end{itemize} + +In der letzten Typisierung stellt $\alpha$ einen beliebigen Typen dar. + +Ein Typkontext $\Gamma$ ordnet jeder freien Variable $x$ einen Typ $\Gamma(x)$ +durch folgende Regeln zu: + +\begin{align*} + \text{\texttt{CONST}}:&\frac{c \in \text{Const}}{\Gamma \vdash c: \tau_c}\\ + \text{\texttt{VAR}}: &\frac{\Gamma(x) = \tau}{\Gamma \vdash c: \tau}\\ + \text{\texttt{ABS}}: &\frac{\Gamma, x: \tau_1 \vdash t: \tau_2}{\Gamma \vdash \lambda x. t: \tau_1 \rightarrow \tau_2}\\ + \text{\texttt{APP}}: &\frac{\Gamma \vdash t_1, \tau_2 \tau\;\;\; \Gamma \vdash t_2: \tau_2}{\Gamma \vdash t_1 t_2: \tau} +\end{align*} + +Dabei ist der lange Strich kein Bruchstrich, sondern ein Symbol der Logik das als +\textbf{Schlussstrich}\xindex{Schlussstrich} bezeichnet wird. Dabei ist der +Zähler als Voraussetzung und der Nenner als Schlussfolgerung zu verstehen. + +\begin{definition}[Typsubstituition]\xindex{Typsubstituition}% + Eine \textit{Typsubstituition} ist eine endliche Abbildung von Typvariablen auf + Typen. +\end{definition} + +Für eine Menge von Typsubsitutionen wird überlicherweise $\sigma$ als Symbol +verwendet. Man schreibt also beispielsweise: +\[\sigma = [\alpha_1 \text{\pointer} \text{\texttt{bool}}, \alpha_2 \text{\pointer} \alpha_1 \rightarrow \alpha_1]\] + +\begin{definition}[Lösung eines Typkontextes] + Sei $t$ eine beliebige freie Variable, $\tau = \tau(t)$ ein beliebiger Typ + $\sigma$ eine Menge von Typsubstitutionen und $\Gamma$ ein Typkontext. + + $(\sigma, \tau)$ heißt eine Lösung für $(\Gamma, t)$, falls gilt: + \[\sigma \Gamma \vdash t : \tau\] +\end{definition} \ No newline at end of file diff --git a/documents/Programmierparadigmen/Vorwort.tex b/documents/Programmierparadigmen/Vorwort.tex index 07afe8a..f114d6a 100644 --- a/documents/Programmierparadigmen/Vorwort.tex +++ b/documents/Programmierparadigmen/Vorwort.tex @@ -1,6 +1,11 @@ +%!TEX root = Programmierparadigmen.tex \chapter*{Vorwort} Dieses Skript wird/wurde im Wintersemester 2013/2014 -von Martin Thoma geschrieben. Das Ziel dieses Skriptes ist vor allem +von Martin Thoma zur Vorlesung von Prof.~Dr.~Snelting geschrieben. Dazu wurden +die Folien von Prof.~Dr.~Snelting benutzt, die Struktur sowie einige Beispiele, +Definitionen und Sätze übernommen. + +Das Ziel dieses Skriptes ist vor allem in der Klausur als Nachschlagewerk zu dienen; es soll jedoch auch vorher schon für die Vorbereitung genutzt werden können und nach der Klausur als Nachschlagewerk dienen. @@ -26,7 +31,7 @@ TODO Grundlegende Kenntnisse vom Programmieren, insbesondere mit Java, wie sie am KIT in \enquote{Programmieren} vermittelt werden, werden vorausgesetzt. Außerdem könnte ein grundlegendes Verständnis für -das O-Kalkül aus \enquote{Grundbegriffe der Informatik} hilfreich sein. +das $\mathcal{O}$-Kalkül aus \enquote{Grundbegriffe der Informatik} hilfreich sein. Die Unifikation wird wohl auch in \enquote{Formale Systeme} erklärt; das könnte also hier von Vorteil sein. diff --git a/documents/Programmierparadigmen/scripts/python/multiplikation.py b/documents/Programmierparadigmen/scripts/python/multiplikation.py new file mode 100644 index 0000000..02fc4f3 --- /dev/null +++ b/documents/Programmierparadigmen/scripts/python/multiplikation.py @@ -0,0 +1,6 @@ +>>> 0.1*0.1*0.3 +0.0030000000000000005 +>>> (0.1*0.1)*0.3 +0.0030000000000000005 +>>> 0.1*(0.1*0.3) +0.003 \ No newline at end of file