From 6eb5b0a00fe94e0c650853ac331375408bf7dfef Mon Sep 17 00:00:00 2001 From: Dominik Stahl Date: Thu, 27 Mar 2025 13:51:53 +0100 Subject: [PATCH] add .gitignore and update image paths in Digitaltechnik; remove unused images from Lineare Algebra; add new content for Theoretische Informatik 1 --- .gitignore | 4 + Digitaltechnik/{ => images}/D_FF.png | Bin Digitaltechnik/{ => images}/JK_FF.png | Bin Digitaltechnik/{ => images}/TPS_RS_FF.png | Bin Digitaltechnik/{ => images}/T_FF.png | Bin Digitaltechnik/inhalt/04_Schaltwerke.tex | 8 +- Lineare Algebra/bijektiv.png | Bin 36534 -> 0 bytes Lineare Algebra/injektiv.png | Bin 35151 -> 0 bytes Lineare Algebra/surjektiv.png | Bin 36359 -> 0 bytes Theoretische Informatik 1/inhalt/01_Logik.tex | 63 +++++++++++++ .../inhalt/02_Relationen.tex | 37 ++++++++ .../inhalt/03_Algorithmen.tex | 84 ++++++++++++++++++ .../inhalt/04_Deklarative_Programmierung.tex | 19 ++++ Theoretische Informatik 1/main.tex | 76 ++++++++++++++++ 14 files changed, 287 insertions(+), 4 deletions(-) create mode 100644 .gitignore rename Digitaltechnik/{ => images}/D_FF.png (100%) rename Digitaltechnik/{ => images}/JK_FF.png (100%) rename Digitaltechnik/{ => images}/TPS_RS_FF.png (100%) rename Digitaltechnik/{ => images}/T_FF.png (100%) delete mode 100644 Lineare Algebra/bijektiv.png delete mode 100644 Lineare Algebra/injektiv.png delete mode 100644 Lineare Algebra/surjektiv.png create mode 100644 Theoretische Informatik 1/inhalt/01_Logik.tex create mode 100644 Theoretische Informatik 1/inhalt/02_Relationen.tex create mode 100644 Theoretische Informatik 1/inhalt/03_Algorithmen.tex create mode 100644 Theoretische Informatik 1/inhalt/04_Deklarative_Programmierung.tex create mode 100644 Theoretische Informatik 1/main.tex diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..76198b5 --- /dev/null +++ b/.gitignore @@ -0,0 +1,4 @@ +*.* +!.gitignore +!*.tex +!*.png \ No newline at end of file diff --git a/Digitaltechnik/D_FF.png b/Digitaltechnik/images/D_FF.png similarity index 100% rename from Digitaltechnik/D_FF.png rename to Digitaltechnik/images/D_FF.png diff --git a/Digitaltechnik/JK_FF.png b/Digitaltechnik/images/JK_FF.png similarity index 100% rename from Digitaltechnik/JK_FF.png rename to Digitaltechnik/images/JK_FF.png diff --git a/Digitaltechnik/TPS_RS_FF.png b/Digitaltechnik/images/TPS_RS_FF.png similarity index 100% rename from Digitaltechnik/TPS_RS_FF.png rename to Digitaltechnik/images/TPS_RS_FF.png diff --git a/Digitaltechnik/T_FF.png b/Digitaltechnik/images/T_FF.png similarity index 100% rename from Digitaltechnik/T_FF.png rename to Digitaltechnik/images/T_FF.png diff --git a/Digitaltechnik/inhalt/04_Schaltwerke.tex b/Digitaltechnik/inhalt/04_Schaltwerke.tex index c15c925..fa9ccb6 100644 --- a/Digitaltechnik/inhalt/04_Schaltwerke.tex +++ b/Digitaltechnik/inhalt/04_Schaltwerke.tex @@ -26,7 +26,7 @@ Aktiv bei steigender oder fallender Flanke. \subsubsection{RS-Flip-Flop} -\includegraphics[width=0.2\textwidth]{TPS_RS_FF.png} +\includegraphics[width=0.2\textwidth]{images/TPS_RS_FF.png} 2 Eingänge: S (Set) und R (Reset) \\ 2 Ausgänge: Q und Q* (Q negiert) \\ @@ -35,7 +35,7 @@ Verbotene Zustände: S = R = 1 \subsubsection{D-Flip-Flop} -\includegraphics[width=0.2\textwidth]{D_FF.png} +\includegraphics[width=0.2\textwidth]{images/D_FF.png} 1 Eingang: D (Data) \\ 2 Ausgänge: Q und Q* (Q negiert) \\ @@ -44,7 +44,7 @@ Keine verbotenen Zustände \subsubsection{JK-Flip-Flop} -\includegraphics[width=0.2\textwidth]{JK_FF.png} +\includegraphics[width=0.2\textwidth]{images/JK_FF.png} 2 Eingänge: J und K \\ 2 Ausgänge: Q und Q* (Q negiert) \\ @@ -53,7 +53,7 @@ Keine verbotenen Zustände (J=K=1: Togglen) \subsubsection{T-Flip-Flop} -\includegraphics[width=0.2\textwidth]{T_FF.png} +\includegraphics[width=0.2\textwidth]{images/T_FF.png} 1 Eingang: T (Toggle) \\ 2 Ausgänge: Q und Q* (Q negiert) \\ diff --git a/Lineare Algebra/bijektiv.png b/Lineare Algebra/bijektiv.png 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z9uiS>Bbmj;{y{G}Ml2=rO)k6qOM_~J5PPBiL@AnMrCp<>L6nG^${iOr2 z9@MaEDM6BXwV#}g{#~p7DJUrDBWwio_}DR(T5VNfgF)>94JybWDVXH;st9h3F6t2D zh#rM0^2!ZtTO0~;H9hHBtxtuQSxFr1lbr708387lnMTcFHZ+4)X00OF07Om=3QN{K zK6O3i79bppA!CW%f$Y*gxH(`>0RIIu<5q6gZf5SAHu*f`jG|f!hmbKi4++Tcm!Dy1 z9S|X~khon@7&!&ZULC8cx| ziDq>y{Ii9C0*?n?M%MD_T<3f&{1o`pL<1KrP5{62S&DI`>d4-J(g6G+SiyMaVYT}G z+_-TXUto}~g0A0wA3ObOzFnju2(W-MD9{4!9_67g^ZYx|8o@6lTf}4q6ig)Llo_c@${Mt zGYl;jA`Z5Zpgr@U@g<95bJ_Mg#Yb4CW_3L18CH@VI44Eor#EjPwlJwyy4d|TvnqB# zCPLJt3@)p?U~qh-3yWrAznNEqD+P{(ZkW}-Dun3d7zimM3z!a`L`5hWzbHOc1OI{P zhv%TJqeDH>&&X1RlnxAxIs_ACDv=)+o}wr)#`DkY1q;B;8wIUws(Hla78fHz$+Y4w zT;RGO?4zc!Pk6QOPLrme$9#RWXn4&es(PR9aX&ZrBe7vxuhT@{lz|Es?d=^IsI(IS2?fKx^5eM4(k=()l^H6BD55(xVKqnQe|4q5ix4 z{Z~UBANy95kzY@RX&1ma6omm|A(urtXo%Wz1eSE=#gz8Kf+GpiiVCHx&Dpm z#FDr4^3szW{!i#eA*5Nq{tNrz>sPkJ+)vT$J(GW?P0Q&=prn2nYz x$ (Alle Elemente, die nur rechts in den Tupeln auftauchen) + +\subsection{Äquivalenzklassen} + +\textbf{Äquivalenzklasse}: $[x] = \{y \in M | x \sim y\}$ (Alle elemente, die $x$ links im Tupel stehen haben) \\ +\textbf{Äquivalenzrelation}: $x \sim y \Leftrightarrow [x] = [y]$ \\ +\textbf{Menge der Äquivalenzklassen}: $R / M = \{[x] | x \in M\}$ (Keine Duplikate) diff --git a/Theoretische Informatik 1/inhalt/03_Algorithmen.tex b/Theoretische Informatik 1/inhalt/03_Algorithmen.tex new file mode 100644 index 0000000..2e2c2ac --- /dev/null +++ b/Theoretische Informatik 1/inhalt/03_Algorithmen.tex @@ -0,0 +1,84 @@ +\section{Algorithmen} + +\subsection{Teilsummenproblem} + +Ist ein NP-vollständiges Problem. + +Jede Zahl darf nur einmal verwendet werden. +In der Summe muss eine bestimmte Zahl erreicht werden. + +\subsection{3-SAT Problem} + +Ist ein NP-vollständiges Problem. + +Finden einer Belegung an Variablen, sodass eine Formel erfüllt wird. + +Die Formel hat die Form: + +\[ + (x_1 \lor x_2 \lor x_3) \land (x_4 \lor x_5 \lor x_6) \land \ldots +\] + +\subsubsection{Umwandlung in Teilsummenproblem} + +Zielzahl: +\[ + 1 \dots 1 \ 4 \dots 4 +\] +mit $1$ für jede Variable und $4$ für jede Klausel (Klammer). + +Zahlen aus denen die Summe gebildet wird: + +\textbf{1. Teil}: 1 an der Stelle der Variable, 0 ansonsten + +\textbf{2. Teil}: 1 an der Stelle an der Die Variable in der Klausen positiv/negativ vorkommt, 0 ansonsten + +\begin{tabular}{c | l | l} + Variable & positive Belegung & negative Belegung \\ + & (1er wenn positiv) & (1er wenn negativ) \\ + \hline + $x_1$ & $0 \dots 001 \ pos_1 \dots pos_n$ & $0 \dots 001 \ neg_1 \dots neg_n$ \\ + $x_2$ & $0 \dots 010 \ pos_1 \dots pos_n$ & $0 \dots 010 \ neg_1 \dots neg_n$ \\ + $x_3$ & $0 \dots 100 \ pos_1 \dots pos_n$ & $0 \dots 100 \ neg_1 \dots neg_n$ \\ + \dots & \dots & \dots +\end{tabular} + +\subsection{Programmablaufplan} + +\subsubsection{Allgemeinheit} +Löst eine Klasse von Problemen. + +\subsubsection{Ausführbarkeit} +Jeder Schritt ist ausführbar. + +\subsubsection{Determinismus} +Jeder Schritt ist eindeutig, unabängig von Ein und Ausgaben. + +\subsubsection{Determiniertheit} +Bei gleichen Eingaben, gleiche Ausgaben. + +\subsubsection{Finitheit} +Beschreibung besitzt endliche Länge. + +\subsubsection{Terminierung} +Die Ausführung endet bei einer endlichen Eingabe + +\subsubsection{Dynamische Finitheit} +Beschränkter Resourcenverbrauch bei endlicher Eingabe. + +\subsubsection{Komplexität} +Die Laufzeit / Resourcenverbrauch ist abschätzbar. + +\subsection{Hoare-Kalkül} + +\subsubsection{Vorbedingung} +Bedingung die vor der Ausführung der Zeile gelten muss. + +\subsubsection{Nachbedingung} +Bedingung die nach der Ausführung der Zeile gelten muss. + +\subsubsection{Schleifenbedingung} +Bedingung die vor jedem Schleifendurchlauf gelten muss. + +\subsubsection{Schleifeninvariante} +Bedingung die vor und nach jedem Schleifendurchlauf gelten muss. diff --git a/Theoretische Informatik 1/inhalt/04_Deklarative_Programmierung.tex b/Theoretische Informatik 1/inhalt/04_Deklarative_Programmierung.tex new file mode 100644 index 0000000..63d3fbf --- /dev/null +++ b/Theoretische Informatik 1/inhalt/04_Deklarative_Programmierung.tex @@ -0,0 +1,19 @@ +\section{Deklarative Programmierung} + +\subsection{Ausdrucksbäume} + +Umwandlung von Ausdrücken in Bäume. + +\subsection{Unifizierung} + +Wahl von Variablen, sodass zwei Ausdrücke gleich sind. + +\subsection{Prolog} + +Umsetzen von Problemen in Prolog. + +\subsection{Haskell} + +Umsetzen von Problemen in Haskell. + +Typen eines Ausdrucks angeben. diff --git a/Theoretische Informatik 1/main.tex b/Theoretische Informatik 1/main.tex new file mode 100644 index 0000000..e342823 --- /dev/null +++ b/Theoretische Informatik 1/main.tex @@ -0,0 +1,76 @@ +\documentclass[10pt,landscape,a4paper]{article} +\usepackage[utf8]{inputenc} +\usepackage[ngerman]{babel} +\usepackage[T1]{fontenc} +%\usepackage[LY1,T1]{fontenc} +%\usepackage{frutigernext} +%\usepackage[lf,minionint]{MinionPro} +\usepackage{tikz} +\usetikzlibrary{shapes,positioning,arrows,fit,calc,graphs,graphs.standard} +\usepackage[nosf]{kpfonts} +\usepackage[t1]{sourcesanspro} +\usepackage{multicol} +\usepackage{wrapfig} +\usepackage[top=4mm,bottom=4mm,left=4mm,right=4mm]{geometry} +\usepackage[framemethod=tikz]{mdframed} +\usepackage{microtype} +\usepackage{pdfpages} + +\let\bar\overline + +\definecolor{myblue}{cmyk}{1,.72,0,.38} + +\def\firstcircle{(0,0) circle (1.5cm)} +\def\secondcircle{(0:2cm) circle (1.5cm)} + +\colorlet{circle edge}{myblue} +\colorlet{circle area}{myblue!5} + +\tikzset{filled/.style={fill=circle area, draw=circle edge, thick}, + outline/.style={draw=circle edge, thick}} + +\pgfdeclarelayer{background} +\pgfsetlayers{background,main} + +\everymath\expandafter{\the\everymath \color{myblue}} +\everydisplay\expandafter{\the\everydisplay \color{myblue}} + +\renewcommand{\baselinestretch}{.8} +\pagestyle{empty} + +\global\mdfdefinestyle{header}{% +linecolor=gray,linewidth=1pt,% +leftmargin=4mm,rightmargin=4mm,skipbelow=4mm,skipabove=4mm, +} + +\newcommand{\header}{ +\begin{mdframed}[style=header] +\footnotesize +\sffamily +Cheat sheet\\ +by~Your~Name,~page~\thepage~of~2 +\end{mdframed} +} + +\makeatletter +\renewcommand{\section}{\@startsection{section}{1}{0mm}% + {.2ex}% + {.2ex}%x + {\color{myblue}\sffamily\small\bfseries}} +\renewcommand{\subsection}{\@startsection{subsection}{1}{0mm}% + {.2ex}% + {.2ex}%x + {\sffamily\bfseries}} + +\makeatother +\setlength{\parindent}{0pt} + +\begin{document} +\small +\begin{multicols*}{3} + \input{inhalt/01_Logik} + \input{inhalt/02_Relationen} + \input{inhalt/03_Algorithmen} + \input{inhalt/04_Deklarative_Programmierung} +\end{multicols*} +\end{document}