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\documentclass[11pt, a4paper, twoside]{article}
\usepackage[
a4paper,
headsep=5mm,
footskip=0mm,
top=12mm,
left=10mm,
right=10mm,
bottom=10mm
]{geometry}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{makecell}
\usepackage{multicol}
\usepackage[noend]{algorithm2e}
\usepackage[utf8]{inputenc}
\usepackage{fancyhdr}
\usepackage{tikz}
\usetikzlibrary{arrows,automata,positioning, graphs, graphdrawing}
\usegdlibrary {trees}
\usepackage{hyperref}
\hypersetup{
colorlinks=true,
linkcolor=blue,
filecolor=magenta,
urlcolor=cyan,
pdftitle={Overleaf Example},
pdfpagemode=FullScreen,
}
\setlength{\algomargin}{0pt}
\begin{document}
\pagestyle{fancy}
\fancyhead{}
\fancyhead[L]{Theoretische Grundlagen der Informatik}
\fancyhead[R]{Gero Beckmann - \url{https://github.com/Geronymos/}}
\fancyfoot{}
\fancyfoot[R]{\thepage}
\newenvironment{definition}[1]{\noindent\textbf{#1:}}{}
\section{Chomsky-Hierarchie}
\hspace*{-.5cm}
\begin{tabular}{ l l l l l }
Chomsky-Typ & Wortproblem & Definition & Bsp & Maschinenmodell \\
Typ 0 &
semi-entscheidbar &
\makecell{$G = (\Sigma, V, S, R)$ \\ $R beliebig$ }&
universelle Sprache &
NTM/DTM akzeptiert L \\
Typ 1 &
NP-Schwer &
\makecell{$u \rightarrow v, |u| \leq |v|$ \\ $u \in V^+, S \notin V$ \\ $S \rightarrow \epsilon$ } &
$L = \{ a^ib^ic^i | i \leq 1 \}$ &
\makecell{NTM mit Platzbedarf n \\ erkennt Wörter der Länge n in L \\ $\Rightarrow NTAPE(n)$ } \\
Typ 2 &
polynomiell &
\makecell{$A \rightarrow v, A \in V$ \\ $v beliebig$} &
$L = \{ a^ib^i | i \leq 1\}$ &
CYK-Alg. erkennt L in polynom. Zeit, Chomsky-NF, NPDA \\
Typ 3 &
linear &
\makecell{$A \rightarrow v, A \in V$ \\ $V \in \epsilon \cup \Sigma \cdot V$} &
$L = \{ a^i | i \leq 1 \}$ &
NEA/DEA erkennt L \\
\end{tabular}
\subsection{Automaten}
DEA $A = (Q, \Sigma, \delta: Q \times \Sigma \rightarrow Q, s \in Q, F \subseteq Q)$ \\
NEA $A = (Q, \Sigma, \delta: Q \times (\Sigma \cup \{ \epsilon \} \rightarrow 2^Q, s \in Q, F \subseteq Q)$
NPDA \\
DPDA \\
DTM \\
NTM \\
\subsection{Pumping-Lemma}
\begin{multicols}{2}
Erfüllt:
\begin{itemize}
\item["$\exists$"] Wähle $n = 2$
\item["$\forall$"] Betrachte beliebiges $w \in L$ mit $|w| > 2$
\item["$\exists$"] Wähle zerlegung $w = uvx$ mit $u = \epsilon, v = aa, x=a^{2(j-1)}$
\item["$\forall$"] Für alle $i \in \mathbb{N}_0: uv^ix = a^{2i}a^2(j-1) = a^{2(i+j-1)} \in L$
\end{itemize}
Widerlegen:
\begin{itemize}
\item["$\exists$"] Wähle $n = 2$
\item["$\forall$"] Betrachte beliebiges $w \in L$ mit $|w| > 2$
\item["$\exists$"] Wähle zerlegung $w = uvx$ mit $u = \epsilon, v = aa, x=a^{2(j-1)}$
\item["$\forall$"] Für alle $i \in \mathbb{N}_0: uv^ix = a^{2i}a^2(j-1) = a^{2(i+j-1)} \in L$
\end{itemize}
\end{multicols}
\begin{multicols}{2}
Potenzmengenkonstuktion NEA $\rightarrow$ DEA
\begin{tabular}{c | c | c}
Zustand & a & b \\
\hline
$\{\underline{s}\}$ & $\{s, q_1\}$ & $\{f\}$ \\
$\{\underline{s}, q_1\}$ & $\{s, q_1\}$ & $\{f, q_2\}$ \\
$\{f\}$ & $\{f\}$ & $\{f\}$ \\
$\{f, q_2\}$ & $\{f\}$ & $\{f, q_1, q_2\}$ \\
$\{f, \underline{s}\}$ & $\{f, s, q_1\}$ & $\{f\}$ \\
$\{f, \underline{s}, q_1\}$ & $\{f, s, q_1\}$ & $\{f, q_2\}$ \\
\end{tabular}
\begin{tikzpicture}[initial text=,shorten >=1pt,node distance=2cm,on grid,auto]
\node[state,initial,accepting] (S) {$S$};
\node[state] (q_1) [right of=S] {$q_1$};
\node[state] (q_2) [right of=q_1] {$q_2$};
\node[state] (f) [below of=q_1] {$f$};
\path[->]
(S) edge [loop above] node {a} ()
(S) edge node [below] {a} (q_1)
(S) edge node [left] {b} (f)
(q_1) edge [bend right] node [above] {a} (S)
(q_1) edge node [below] {b} (q_2)
(q_2) edge [bend right] node [above] {b} (q_1)
(q_2) edge [loop right] node {b} ()
(q_2) edge node {a} (f)
(f) edge [loop left] node {a,b} ()
;
\end{tikzpicture}
\end{multicols}
\begin{multicols}{2}
Entfernen von $\epsilon$-Übergängen
\begin{tabular}{c | c | c}
Zustand & a & b \\
\hline
$S$ & $q_1$ & $S, q_1, q_2, q_3$ \\
$q_1$ & $q_2, q_3$ & $q_3$ \\
$q_2$ & $q_1$ & $S, q_2, q_3$ \\
$q_3$ & $q_1$ & $S, q_2, q_3$ \\
\end{tabular}
\begin{tikzpicture}[initial text=,shorten >=1pt,node distance=2cm,on grid,auto]
\node[state,initial] (S) {$S$};
\node[state,accepting] (q_1) [right of=S] {$q_1$};
\node[state,accepting] (q_2) [below of=q_1] {$q_2$};
\node[state] (q_3) [below of=S] {$q_3$};
\path[->]
(S) edge node {b} (q_1)
(S) edge node [above left] {$\epsilon$} (q_2)
(q_1) edge node {a} (q_2)
(q_1) edge [bend left] node [above right] {b} (q_3)
(q_2) edge node {\epsilon} (q_3)
(q_3) edge node [below left] {a} (q_1)
(q_3) edge node {b} (S)
(q_3) edge [loop left] node {b} ()
;
\end{tikzpicture}
\end{multicols}
Minimierung von Automaten
\begin{tikzpicture}[initial text=,shorten >=1pt,node distance=2cm,on grid,auto]
\node[state,initial] (S) {$S$};
\node[state] (p) [right of=S] {$p$};
\node[state] (q) [right of=p] {$q$};
\node[state] (t) [below of=p] {$t$};
\node[state,accepting] (r) [below of=q] {$r$};
\node[state] (v) [below of=t] {$v$};
\node[state] (u) [below of=r] {$u$};
\path[->]
(S) edge [loop above] node {0} ()
(S) edge node {1} (p)
(p) edge [loop above] node {1} ()
(p) edge node {0} (q)
(q) edge [bend left] node {0} (S)
(q) edge node {1} (r)
(t) edge node [right] {0} (S)
(t) edge [bend right] node {1} (r)
(r) edge [bend right] node [above] {0} (t)
(r) edge node {1} (u)
(v) edge node {0} (S)
(v) edge node[left] {1} (r)
(u) edge node {0} (v)
(u) edge [loop right] node {1} ()
;
\end{tikzpicture}
\begin{tikzpicture}[initial text=,shorten >=1pt,node distance=2cm,on grid,auto]
\node[state,initial] (S) {$[S]$};
\node[state] (p) [right of=S] {$[p]$};
\node[state] (q) [right of=p] {$[q]$};
\node[state,accepting] (r) [right of=q] {$[r]$};
\path[->]
(S) edge [loop above] node {0} ()
(S) edge node {1} (p)
(p) edge [loop above] node {1} (p)
(p) edge node {0} (q)
(q) edge[bend left] node {0} (S)
(q) edge node {1} (r)
(r) edge[bend right] node [above] {1} (p)
;
\end{tikzpicture}
\begin{tikzpicture} [binary tree layout]
\node[align=center] (1) {s,p,q,r,t,a,v \\ $\epsilon$ trennt}
child {
node {r}
}
child { node[align=center] {s,p,q,t,u,v \\ 1 trennt}
child { node[align=center] {s,p,u \\ 0 trennt}
child { node {s} }
child { node {p,u} }
}
child{
node {q,t,v}
}
};
\end{tikzpicture}
\subsection{Nerode-Relation}
\subsection{Chomsky-NF}
\section{NP-Vollständigkeit}
\section{Kellerautomaten}
\subsection{$4COLOR \in NP$}
\subsection{$3COLOR \propto 4COLOR$}
\subsubsection{Transformation}
\subsubsection{Äquivalenz/Korrektheit}
\section{Approximationsalgorithmen}
\section{Huffman-Kodierung}
\end{document}
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