From 28e588ba5c30e6417d876490a3007cc331c48fb5 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 3 Oct 2025 19:42:49 +0200 Subject: [PATCH] [TI] Continue summary on finite automata --- .../02_finite-automata/00_representation.tex | 40 +++++++++++++++--- semester3/ti/ti-summary.pdf | Bin 828624 -> 834892 bytes 2 files changed, 35 insertions(+), 5 deletions(-) diff --git a/semester3/ti/parts/02_finite-automata/00_representation.tex b/semester3/ti/parts/02_finite-automata/00_representation.tex index bfefdd9..a31ea76 100644 --- a/semester3/ti/parts/02_finite-automata/00_representation.tex +++ b/semester3/ti/parts/02_finite-automata/00_representation.tex @@ -55,14 +55,44 @@ Heute verwendet man meist einen gerichteten Graphen $G(A)$: \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, $C_i \vdash_M C_{i + 1}$. + \item \bi{Schritt}: Relation auf Konfigurationen $\bigvdash{M}{} \subseteq (Q \times \Sigma^*) \times (Q \times \Sigma^*$ definiert durch $(q, w) \bigvdash{M}{} (p, x) \Leftrightarrow w = ax, a \in \Sigma \text{ und } \delta(q, a) = p$. Einfacher: Anwendung von $\delta$ auf die aktuelle Konfiguration + \item \bi{Berechnung} $C$: Endliche Folge von Konfigurationen, $C_i \bigvdash{M}{} C_{i + 1}$. 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 + \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} +Die Übergangsfunktion kann auch gut graphisch oder tabellarisch (wie eine Truth-Table) dargestellt werden. -% P68 +\begin{definition}[]{Reflexive und transitive Hülle} + Sei $M = (Q, \Sigma, \delta, q_0, F)$ ein endlicher Automat. Die reflexive und transitive Hülle $\bigvdash{M}{*}$ der Schrittrelation $\bigvdash{M}{}$ von $M$ als + $(q, w) \bigvdash{M}{*} (p, u) \Leftrightarrow (q = p \land w = u) \lor \exists k \in \N - \{ 0 \}$ so dass + \begin{enumerate}[label=\textit{(\roman*)}] + \item $w = a_1\ldots a_ku, a_i \in \Sigma$ für $i = 1, \ldots, k$ + \item $\exists r_1, \ldots, r_{k - 1} \in Q$, so dass + $(q, w) \bigvdash{M}{} (r_1, a_2 \ldots a_k u) \bigvdash{M}{} (r_2, a_3 \ldots a_k u) \bigvdash{M}{} \ldots (r_{k - 1}, a_k u) \bigvdash{M}{} (p, u)$ + \end{enumerate} + Wir definieren $\hat{\delta}: Q \times \Sigma^* \rightarrow Q$ durch + \rmvspace + \rmvspace + \begin{multicols}{2} + \begin{enumerate}[label=\textit{(\roman*)}] + \item $\hat{\delta}(q, \lambda) = q \smallhspace \forall q \in Q$ + \item $\hat{\delta}(q, wa) = \delta(\hat{\delta}(q, w), a) \forall a \in \Sigma, w \in \Sigma^*, q \in Q$ + \end{enumerate} + \end{multicols} +\end{definition} +Und $(q, w) \bigvdash{M}{*} (p, u)$ bedeutet, dass es eine Berechnung von $M$ gibt, die von der Konfiguration $(q, w)$ zu $(p, u)$ führt, +während $\hat{\delta}(q, w) = p$ bedeutet einfach $(q, w) \bigvdash{M}{*} (p, \lambda)$, also falls $M$ im Zustand $q$ das Wort $w$ zu lesen beginnt, $M$ im Zustand $p$ endet. +Also gilt $L(M) = \{ w \in \Sigma^* \divides (q_0, w) \bigvdash{M}{*} (p, \lambda) \text{ mit } p \in F \} = \{ w \in \Sigma^* \divides \hat{\delta}(q_0, w) \in F \}$ + +\inlinelemma $L(M) = \{ w \in \{ 0, 1 \}^* \divides |w|_0 + |w|_1 \equiv 0 \text{ mod } 2 \}$ + +Jeder EA teilt die Menge $\Sigma^*$ in $|Q|$ Klassen +$\text{Kl}[p] = \{ w \in \Sigma^* \divides \hat{\delta}(q_0, w) = p \} = \{ w \in \Sigma^* \divides (q_0, w) \bigvdash{M}{*} (p, \lambda) \}$ +und entsprechend $\bigcup_{p \in Q} \text{Kl}[p] = \Sigma^*$ und $\text{Kl}[p] \text{Kl}[q] = \emptyset \smallhspace \forall p \neq q \in Q$. +In dieser Terminologie gilt dann $L(M) = \bigcup_{p \in F} \text{Kl}[p]$ + +Wir können $L(M)$ mit Klassen bestimmen und haben eine 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