From 5f416d93038d980cc078d1f231a9115f6f1e9869 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 3 Oct 2025 18:36:52 +0200 Subject: [PATCH] [TI] Continue for finite automaton --- .../02_finite-automata/00_representation.tex | 41 ++++++++++++++++++ semester3/ti/ti-summary.pdf | Bin 821015 -> 828624 bytes 2 files changed, 41 insertions(+) diff --git a/semester3/ti/parts/02_finite-automata/00_representation.tex b/semester3/ti/parts/02_finite-automata/00_representation.tex index bd03bc8..bfefdd9 100644 --- a/semester3/ti/parts/02_finite-automata/00_representation.tex +++ b/semester3/ti/parts/02_finite-automata/00_representation.tex @@ -25,3 +25,44 @@ 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 und die gelesene Eingabe dann dem \bi{Programm} weitergibt. + +Diese Notation wird jedoch heute kaum mehr verwendet (because \texttt{goto} bad, Prof. Roscoe would approve). +Heute verwendet man meist einen gerichteten Graphen $G(A)$: +\begin{itemize} + \item Hat so viele Knoten (= \bi{Zustände}) wie das Programm $A$ Zeilen hat + \item Wenn das Programm beim Lesen von Symbol $b$ von Zeile $i$ auf $j$ sprint, so gibt es in $G(A)$ eine gerichtete Kante $(i, j)$ von Knoten $i$ nach Knoten $j$ mit Markierung $b$. Sie wird als \bi{Übergangsfunktion} bezeichnet + \item Jeder Knoten hat den Ausgangsgrad $|\Sigma|$ (wir müssen alle Fälle abdecken) +\end{itemize} + +\begin{definition}[]{Endlicher Automat} + Ist eine Quitupel $M = (Q, \Sigma, \delta, q_0, F)$: + \begin{enumerate}[label=\textit{(\roman*)}] + \item $Q$ ist eine endliche Menge von \bi{Zuständen} + \item $\Sigma$ ist das \bi{Eingabealphabet} + \item $\delta: Q \times \Sigma \rightarrow Q$ ist die \bi{Übergangsfunktion}. $\delta(q, a) = p$ bedeutet Übergang von Zustand $q$ nach $p$ falls in $q$ $a$ gelesen wurde + \item $q_0 \in Q$ ist der \bi{Anfangszustand} + \item $F \subseteq Q$ ist die \bi{Menge der akzeptierenden Zustände} + \end{enumerate} + \rmvspace + \rmvspace + \begin{multicols}{2} + \begin{itemize} + \item \bi{Konfiguration}: Element aus $Q \times \Sigma^*$ + \item \bi{Startkonfiguration} auf $x$: $(q_0, x)$ + \item \bi{Endkonfiguration}: Jede aus $Q \times \{ \lambda \}$ + \end{itemize} + \end{multicols} + \rmvspace + \rmvspace + \begin{itemize} + \item \bi{Schritt}: Relation auf Konfigurationen $\vdash_M \subseteq (Q \times \Sigma^*) \times (Q \times \Sigma^*$ definiert durch $(q, w) \vdash_M (p, x) \Leftrightarrow w = ax, a \in \Sigma \text{ und } \delta(q, a) = p$. Einfacher: Anwendung von $\delta$ auf die aktuelle Konfiguration + \item \bi{Berechnung} $C$: Endliche Folge von Konfigurationen, $C_i \vdash_M C_{i + 1}$. + Auf Eingabe $x \in \Sigma^*$, $C_0$ Startkonfiguration und $C_n$ Endkonfiguration. + Falls $C_n \in F \times \{ \lambda \}$, $C$ \bi{akzeptierende Berechnung}, $M$ \bi{akzeptiert Wort} $x$. + Anderenfalls ist $C$ eine \bi{verwerfende Berechnung} und $M$ \bi{verwirft (akzeptiert nicht) das Wort} $x$ + \item \bi{Akzeptierte Sprache} $L(M) = \{ w \in \Sigma^* \divides \text{$M$ akzeptiert das Wort } w \text{ und $M$ endet in Endkonfig.} \}$ + \item $\mathcal{L}_{EA} = \{ L(M) | M \text{ ist ein EA}\}$ ist die Klasse aller Sprachen die von endlichen Automaten akzeptiert werden, auch genannt \bi{Klasse der regulären Sprachen} und für jede Sprache $L \in \mathcal{L}_{EA}$ gilt: $L$ regulär + \end{itemize} +\end{definition} + +% P68 diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 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