eth-summaries/semester3/ti/main/ti-summary.tex

\documentclass{article}

\input{~/projects/latex/dist/full.tex}
\setLang{de}

\setup{Theoretische Informatik}

\newcommand{\hdelta}{\hat{\delta}}
\newcommand{\qacc}{q_{\text{accept}}}
\newcommand{\qrej}{q_{\text{reject}}}
\newcommand{\ldiag}{L_{\text{diag}}}
\newcommand{\lempty}{L_{\text{empty}}}
\renewcommand{\tc}{\text{Time}}
\newcommand{\spc}{\text{Space}}
\usetikzlibrary{automata, positioning, arrows.meta}

\begin{document}
\startDocument
\usetcolorboxes

\vspace{1cm}
\begin{center}
    \begin{tikzpicture}[node distance = 1cm and 2cm, >={Stealth[round]}]
        \node[state, initial left, accepting] (q0p0) {$q_0, p_0$};
        \node[state] (q0p1) [right=of q0p0] {$q_0, p_1$};
        \node[state] (q0p2) [right=of q0p1] {$q_0, p_2$};
        \node[state, accepting] (q1p0) [below=of q0p0] {$q_1, p_0$};
        \node[state] (q1p1) [right=of q1p0] {$q_1, p_1$};
        \node[state] (q1p2) [right=of q1p1] {$q_1, p_2$};
        \node[state, accepting] (q2p0) [below=of q1p0] {$q_2, p_0$};
        \node[state, accepting] (q2p1) [right=of q2p0] {$q_2, p_1$};
        \node[state, accepting] (q2p2) [right=of q2p1] {$q_2, p_2$};

        \path[->]
        % Level 0
        (q0p0) edge node [above] {a} (q0p1)
        (q0p1) edge node [above] {a} (q0p2)
        (q0p2) edge [bend right] node [above] {a} (q0p0)
        % Level 0 to level 1
        (q0p0) edge node [right] {b} (q1p0)
        (q0p1) edge node [right] {b} (q1p1)
        (q0p2) edge node [right] {b} (q1p2)
        % Level 1 to level 2
        (q1p0) edge node [above] {a} (q2p1)
        (q1p1) edge node [above] {a} (q2p2)
        (q1p2) edge node [right, xshift=0.3cm] {a} (q2p0)
        % Level 2 to level 1
        (q2p0) edge node [right] {b} (q1p0)
        (q2p1) edge node [above left, yshift=0.1cm] {b} (q1p1)
        (q2p2) edge node [right] {b} (q1p2)
        % Level 2
        (q2p0) edge node [above] {a} (q2p1)
        (q2p1) edge node [above] {a} (q2p2)
        (q2p2) edge [bend left] node [below] {a} (q2p0)
        % ────────────────────────────────────────────────────────────────────
        % Loops on level 1
        (q1p0) edge [loop left] node {b} ()
        (q1p1) edge [loop left] node {b} ()
        (q1p2) edge [loop left] node {b} ();
    \end{tikzpicture}
\end{center}

\vspace{3cm}
\begin{center}
    \begin{Large}
        ``\textit{Sie können also alle C Programme in Kanonischer Ordnung aufzählen. Sollten Sie dies tun? Wahrscheinlich nicht. Was aber zählt ist, sie \textbf{können} es tun}''
    \end{Large}

    \hspace{3cm} - Prof. Dr. Dennis Komm, 2025
\end{center}

\vspace{3cm}
\begin{center}
    HS2025, ETHZ\\[0.2cm]
    \begin{Large}
        Summary of the book \color{MidnightBlue}\fbox{\href{https://link.springer.com/book/10.1007/978-3-658-06433-4}{Theoretische Informatik}}\color{black}
    \end{Large}\\[0.2cm]
    by Prof. Dr. Juraj Hromkovic
\end{center}

\newpage

\printtoc{Orange}

\begin{scriptsize}
    \begin{itemize}
        \item \textit{Note: Definitions, Lemmas, etc are often 1:1 copies from the book or paraphrased (as I did not find an easier way of stating them)}
        \item \textit{Note: In case I forgot to add the PDF page numbers, the PDF page number is given by $P_{\text{PDF}} = P_{\text{Book}} + 15$}
    \end{itemize}
\end{scriptsize}

\newpage


% Combinatorics
\input{parts/combinatorics.tex}


% ┌                                                ┐
% │                   Alphabets                    │
% └                                                ┘
\newsection
\section{Alphabete, Wörter, Sprachen und Darstellung von Problemen}
\stepcounter{subsection}
\input{parts/01_languages-problems/00_alphabet.tex}
\input{parts/01_languages-problems/01_algorithmic-problems.tex}
\input{parts/01_languages-problems/02_kolmogorov-complexity.tex}


% ────────────────────────────────────────────────────────────────────
% ┌                                                ┐
% │                    Automata                    │
% └                                                ┘
\newsection
\section{Endliche Automaten}
\stepcounter{subsection}
\input{parts/02_finite-automata/00_representation.tex}
\input{parts/02_finite-automata/01_simulations.tex}
\input{parts/02_finite-automata/02_proofs-of-nonexistance.tex}
\input{parts/02_finite-automata/03_non-determinism.tex}

\newsection
\section{Turing-Maschinen}
\setcounter{subsection}{2}
\input{parts/03_turing_machines/00_intro.tex}
\input{parts/03_turing_machines/01_multi-band-church-thesis.tex}
\input{parts/03_turing_machines/02_non-deterministic.tex}

\newsection
\section{Berechenbarkeit}
\stepcounter{subsection}
\input{parts/04_computability/00_intro.tex}
\input{parts/04_computability/01_reduction.tex}
\input{parts/04_computability/02_rice.tex}
\input{parts/04_computability/03_kolmogorov.tex}

\newsection
\section{Komplexitätstheorie}
\stepcounter{subsection}
\input{parts/05_complexity/00_intro.tex}
\input{parts/05_complexity/01_class_p.tex}
\input{parts/05_complexity/02_non-deterministic-complexity.tex}
\input{parts/05_complexity/03_class-np.tex}
\input{parts/05_complexity/04_np-completeness.tex}

\newsection
\section{Grammatiken}
\stepcounter{subsection}
\input{parts/06_grammars/00_intro.tex}


\end{document}