From b73c28734187cf2aff3c57912d79106aa7c52660 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 2 Oct 2025 11:21:34 +0200 Subject: [PATCH] [TI] Introduction to automata --- .../00_alphabet.tex} | 0 .../01_algorithmic-problems.tex} | 0 .../02_kolmogorov-complexity.tex} | 0 .../02_finite-automata/00_representation.tex | 27 ++++++++++++++++++ semester3/ti/ti-summary.pdf | Bin 817733 -> 821015 bytes semester3/ti/ti-summary.tex | 18 ++++++++++-- 6 files changed, 42 insertions(+), 3 deletions(-) rename semester3/ti/parts/{languages-problems/alphabet.tex => 01_languages-problems/00_alphabet.tex} (100%) rename semester3/ti/parts/{languages-problems/algorithmic-problems.tex => 01_languages-problems/01_algorithmic-problems.tex} (100%) rename semester3/ti/parts/{languages-problems/kolmogorov-complexity.tex => 01_languages-problems/02_kolmogorov-complexity.tex} (100%) create mode 100644 semester3/ti/parts/02_finite-automata/00_representation.tex diff --git a/semester3/ti/parts/languages-problems/alphabet.tex b/semester3/ti/parts/01_languages-problems/00_alphabet.tex similarity index 100% rename from semester3/ti/parts/languages-problems/alphabet.tex rename to semester3/ti/parts/01_languages-problems/00_alphabet.tex diff --git a/semester3/ti/parts/languages-problems/algorithmic-problems.tex b/semester3/ti/parts/01_languages-problems/01_algorithmic-problems.tex similarity index 100% rename from semester3/ti/parts/languages-problems/algorithmic-problems.tex rename to semester3/ti/parts/01_languages-problems/01_algorithmic-problems.tex diff --git a/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex b/semester3/ti/parts/01_languages-problems/02_kolmogorov-complexity.tex similarity index 100% rename from semester3/ti/parts/languages-problems/kolmogorov-complexity.tex rename to semester3/ti/parts/01_languages-problems/02_kolmogorov-complexity.tex diff --git a/semester3/ti/parts/02_finite-automata/00_representation.tex b/semester3/ti/parts/02_finite-automata/00_representation.tex new file mode 100644 index 0000000..bd03bc8 --- /dev/null +++ b/semester3/ti/parts/02_finite-automata/00_representation.tex @@ -0,0 +1,27 @@ +\subsection{Darstellung} +Folgende Fragen müssen zur Definition eines Berechnungsmodells beantwortet werden: +\begin{enumerate} + \item Welche elementaren Operationen stehen zur Verfügung (um das Programm zusammenzustellen)? + \item Wie funktioniert der Speicher? + \item Wie funktioniert die Eingabe (und welches Alphabet verwendet sie)? + \item Wie funktioniert die Ausgabe (und welches Alphabet verwendet sie)? +\end{enumerate} +Endliche Automaten haben keinen Speicher, mit Ausnahme des Zeigers (can be understood similarly to a program counter) + +Ein endlicher Automat mit dem Eingabealphabet $\Sigma = \{ a_1, \ldots, a_k \}$ darf nur den Operationstyp \texttt{select} verwenden. +\begin{align*} + \texttt{select } input & = a_1 \texttt{ goto } i_1 \\[-0.2cm] + \vdots \\ + input & = a_k \texttt{ goto } i_k +\end{align*} +Alternativ, falls $|\Sigma| = 2$ (typischerweise für $\alphabets{bool}$), kann man statt \texttt{select} auch \texttt{if\dots then\dots else} nutzen. +Typischerweise werden solche Programme für Entscheidungsprobleme genutzt und die Checks sind dann: +\begin{align*} + \texttt{if } input = 1 \texttt{ then goto } i \texttt{ else goto } j +\end{align*} +Wir wählen eine Teilmenge $F \subseteq \{ 0, \ldots, m - 1 \}$, wobei $m$ die Anzahl Zeilen des Programms ist. +Ist die Zeile auf der das Programm endet ein Element von $F$, so akzeptiert das Programm die Eingabe. +Die Menge $F$ wird auch die \bi{vom Programm akzeptierte Sprache} genannt +% +Ein Programm $A$ arbeitet dann Buchstabe für Buchstabe das Eingabewort ab und springt so also kontinuierlich durch das Programm bis die Eingabe endet. +Mit formaleren Begriffen ist das Eingabewort als \bi{Band} dargestellt, welches von einem \bi{Lesekopf}, der sich nur nach links oder rechts bewegen kann gelesen wird 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Alphabets │ +% └ ┘ \newsection \section{Alphabete, Wörter, Sprachen und Darstellung von Problemen} -\input{parts/languages-problems/alphabet.tex} -\input{parts/languages-problems/algorithmic-problems.tex} -\input{parts/languages-problems/kolmogorov-complexity.tex} +\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} \end{document}