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z>_k6r1Q^kZM=8(%=z@{?1;FN*86+;R~s>uPgdI4R^RT1$&&v7B-s3>7}0>VRd zrpsGcN=}fD%i276$qqvHQf`QOt4s-QmOneSN0gP diff --git a/Automaten, Sprachen und Komplexität - MindMap.tex b/Automaten, Sprachen und Komplexität - MindMap.tex index 7a637f0..8ac864b 100644 --- a/Automaten, Sprachen und Komplexität - MindMap.tex +++ b/Automaten, Sprachen und Komplexität - MindMap.tex @@ -343,9 +343,9 @@ \item kurz: DFA (deterministic finite automaton) \end{itemize*} - \section{Turingmaschine} - Definition: Eine Turingmaschine (TM) ist ein 7-Tupel $$M=(Z,\sum, \Phi, \delta, z_o, \Box, E)$$, wobei + \section{Turingmaschine (TM)} \begin{itemize*} + \item 7-Tupel $M=(Z,\sum, \Phi, \delta, z_o, \Box, E)$ \item $\sum$ das Eingabealphabet \item $\Phi$ mit $\Phi\supseteq\sum$ und $\Phi\cap Z\not= 0$ das Arbeits- oder Bandalphabet, \item $z_0\in Z$ der Startzustand, @@ -354,9 +354,8 @@ \item $E\subseteq Z$ die Menge der Endzustände ist \end{itemize*} - \section{Linksableitung} - Eine Ableitung $S=w_0\Rightarrow w_1\Rightarrow ...\Rightarrow w_r = w\in\sum^*$ heißt Linksableitung, wenn in jedem Schritt das am weitesten links stehende Nichtterminal ersetzt wird (Analog Rechtsableitung). + Eine Ableitung $S=w_0\Rightarrow ...\Rightarrow w_n = w\in\sum^*$ heißt Linksableitung, wenn in jedem Schritt das am weitesten links stehende Nichtterminal ersetzt wird (analog Rechtsabl.) \section{CYK-Algorithmus} Gehört ein gegebenes Wort zu $L(G)$? @@ -368,10 +367,11 @@ \section{Kellerautomaten} Um ein Automatenmodell für Kontextfreie Sprachen zu erhalten führt man einen Keller-(Pushdown)-Speicher ein, auf dem sich eine beliebig lange Sequenz von Zeichen befinden darf. Beim Einlesen eines neuen Zeichens wird das oberste Zeichen des Kellers gelesen und durch eine (evtl. leere) Sequenz von Zeichen ersetzt. An anderen Stellen kann der Keller nicht gelesen/geändert werden - \section{die Greibach-Normalform} + \columnbreak + \section{Greibach-Normalform} Eine kontextfreie Grammatik G ist in Greibach Normalform falls alle Produktionen aus P folgende Form haben: $A\rightarrow aB_1B_2...B_k$, mit $k\in \mathbb{N}, A,B_1,...,B_k\in V$ und $a\in \sum$. Die Greibach Normalform garantiert, dass bei jedem Ableitungsschritt genau ein Alphabetsymbol entsteht. - \section{das Lemma von Ogden (William Ogden)} + \section{Lemma von Ogden (William Ogden)} Wenn L eine kontextfreie Sprache ist, dann gibt es $n\geq 1$ derart, dass für alle $z\in L$, in denen $n$ Positionen markiert sind, gilt: es gibt Wörter $u,v,w,x,y\in\sum^*$ mit \begin{enumerate*} \item $z=uvwxy$ @@ -389,10 +389,11 @@ Sei $R$ die Klasse aller Turing-berechenbaren Funktionen $\{0,1\}^*\rightarrow\{0,1\}^*$, $\Omega$ die nirgendwo definierte Funktion und sei $S\subseteq \mathbb{R}$ mit $\Omega\in S$ und $\not = \mathbb{R}$. Dann ist die Sprache $C(S)=\{w\in L_{TM} | \phi_w\in S\}$ unentscheidbar. \section{Semi Entscheidbarkeit} - Auch wenn das Halteproblem bei leerer Eingabe $H_0$ unentscheidbar ist, so kann doch nach endlicher Zeit festgestellt werden, daß die Maschine $M_w$ bei leerer Eingabe anhält - $H_0$ ist also "halb-" oder "semi-entscheidbar". + Auch wenn das Halteproblem bei leerer Eingabe $H_0$ unentscheidbar ist, so kann doch nach endlicher Zeit festgestellt werden, daß die Maschine $M_w$ bei leerer Eingabe anhält - $H_0$ ist also ,,halb-'' oder ,,semi-entscheidbar''. - Eine Sprache $L\subseteq \sum^*$ heißt semi-entscheidbar, falls die "halbe" charakteristische Funktion von L, d.h. die partielle Funktion $X'_L:\sum^*\rightarrow \{1\}$ mit $x'_L=\begin{cases} 1 \quad\text{ falls } w\in L\\ undef. \quad\text{ falls } w\not\in L \end{cases}$ berechenbar ist. + Eine Sprache $L\subseteq \sum^*$ heißt semi-entscheidbar, falls die ,,halbe'' charakteristische Funktion von L, d.h. die partielle Funktion $X'_L:\sum^*\rightarrow \{1\}$ mit $x'_L=\begin{cases} 1 \quad\text{ falls } w\in L\\ undef. \quad\text{ falls } w\not\in L \end{cases}$ berechenbar ist. + \columnbreak \section{Universelle Turing Maschine} eine Turing-Maschine, die jede Turing-Maschine simulieren kann, wenn deren Kodierung gegeben ist. Buchstaben des Bandalphabets als Wörter über $\{0, 1, 2\}$ mit $\Box = 2$ kodiert. Ab jetzt nehmen wir an, daß wir immer dieses Bandalphabet haben.