From 43b6a6894042741a880f34d384d726d16a85dc1f Mon Sep 17 00:00:00 2001 From: Robert Jeutter Date: Thu, 25 Feb 2021 15:55:21 +0100 Subject: [PATCH] Grundbegriffe & Sortierung --- ... 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{wenn es $x\in\sum^*$ gibt mit $xy=w$}} } } - child{node [subtopic]{Wort} - child[description, level distance=1cm] { node { y Präfix von w, wenn es $z\in\sum^*$ gibt mit $yz=w$}} - child[description, level distance=2cm] { node { y Infix/Faktor von w, wenn es $x,z\in\sum^*$ gibt mit $xyz = w$}} - child[description, level distance=3cm] { node { y Suffix von w, wenn es $x\in\sum^*$ gibt mit $xy=w$}} + child{node [subtopic]{Sprache} + child [theme, level distance=1cm] { node {Chomsky Hierachie} + child[description, level distance=1cm] { node {Typ 0: Allgemein \\ jede Grammatik ist vom Typ 0}} + child[description, level distance=2.5cm] { node {Typ 1: Kontextsensitiv \\ wenn es Wörter $u,v,w\in(V\cup\sum)^*,|v|>0$ und ein Nichtterminal $A\in V$ gibt mit $l=uAw$ und $r=uvw$}} + child[description, level distance=4cm] { node {Typ 2: Kontextfrei \\ wenn $l\in V$ und $r\in (V\cup \sum)^*$ gilt}} + child[description, level distance=5.2cm] { node {Typ 3: Regulär \\ wenn $l\in V$ und $r\in \sum V\cup {\epsilon}$ gilt}} + } + child[theme, level distance=7.4cm]{node {Kleene Abschluss $L*=\bigcup_{n\geq 0} L^n$} + %child[description, level distance=1cm]{node{$L*=\bigcup_{n\geq 0} L^n$}} + } } child{node [subtopic]{Grammatik} child[theme, level distance=1cm] { node {Symbole} - child[description, level distance=1cm] { node {Nicht-Terminale (oder Variablen), aus denen noch weitere Wortbestandteile abgeleitet werden sollen}} - child[description, level distance=2cm] { node {Terminale (die "eigentlichen" Symbole)}} + child[description, level distance=1cm] { node {Nicht-Terminale, Großbuchstaben, Elemente aus V}} + child[description, level distance=2cm] { node {Terminale, Kleinbuchstaben, Elemente aus $\sum$}} } child[theme, level distance=4cm]{ node {4-Tupel $G=(V,\sum, P, S)$} child[description, level distance=1cm] { node {V ist eine endliche Menge von Nicht-Terminalen oder Variablen}} - child[description, level distance=2cm] { node {$\sum$ ist ein Alphabet (Menge der Terminale)}} + child[description, level distance=2cm] { node {$\sum$ ist Alphabet (Menge der Terminale)}} child[description, level distance=3cm] { node {$P$ ist eine endliche Menge von Regeln oder Produktionen}} child[description, level distance=4cm] { node {$S\in V$ ist das Startsymbol oder das Axiom}} } - child[theme, level distance=8cm]{ node {Konventionen} - child[description, level distance=1cm] { node {Variablen sind Großbuchstaben (Elemente aus V)}} - child[description, level distance=2cm] { node {Terminale sind Kleinbuchstaben (Elemente aus $\sum$)}} - } }; \end{tikzpicture} \begin{tikzpicture}[ + topic/.style={ + text centered, + text width=6cm, + level distance=1mm, + sibling distance=5mm, + rounded corners=2pt + }, subtopic/.style={ yshift=1.5cm, text centered, @@ -173,161 +175,251 @@ rounded corners=2pt, fill=gray!10 }, - level 1/.style={sibling distance=5.5cm}, + theme/.style={ + grow=down, + xshift=-0.6cm, + text centered, + text width=5cm, + edge from parent path={(\tikzparentnode.205) |- (\tikzchildnode.west)} + }, + description/.style={ + grow=down, + xshift=-2cm, + text width=5cm, + right, + text centered, + edge from parent path={(\tikzparentnode.189) |- (\tikzchildnode.west)} + }, + level 1/.style={sibling distance=7cm}, level 1/.append style={level distance=2.5cm}, ] % Topic - \node[topic]{Automaten, Sprachen \& Komplexität} - child{node [subtopic] {intuitiv berechenbar} - child[theme, level distance=1cm]{node{$\mu$ rekurisv}} - child[theme, level distance=2cm]{node{while berechnenbar}} - child[theme, level distance=3cm]{node{Turing berechenbar}} - child[theme, level distance=4cm]{node{goto berechnenbar}} + \node[topic]{intuitiv (loop) berechenbar} + child{node [subtopic]{$\mu$ rekurisv} + child[theme, level distance=1cm]{node{Definition} + child[description, level distance=1cm]{node{$\mu f:\mathbb{N}^k\rightarrow\mathbb{N}$ definiert durch }} + child[description, level distance=2cm]{node{$\mu f(n_1,...,n_k)= min\{m| f(m,n_1,...,n_k)=0$ und }} + child[description, level distance=3cm]{node{$\forall x< m: f(x,n_1,...,n_k) \text{ definiert } \}$.}} + } + } + child{node [subtopic]{while berechnenbar} + child[theme, level distance=1cm]{node{Definition} + child[description, level distance=1.2cm]{node{$x_i=c; x_i=x_j+c; x_i=x_j-c$ mit $c\in\{0,1\}$ und $i,j\geq 1$ (Wertzuweisung) oder }} + child[description, level distance=2.5cm]{node{$P_1;P_2$, wobei $P_1$ und $P_2$ bereits While Programme sind oder }} + child[description, level distance=4cm]{node{while $x_i\not = 0$ do P end, wobei P ein While Programm ist und $i\geq 1$. }} + } + child[theme, level distance=6.5cm]{node {Gödel Vermutung}} + } + child{node [subtopic]{goto berechnenbar} + child[theme, level distance=1cm]{node{endliche nichtleere File $P=A_1;A_2;...;A_m$ von Anweisungen $A_i$ der Form} + child[description, level distance=1.5cm]{node{$x_i=c, x_i=x_j+c, x_i=x_j-c$ mit $c\in\{0,1\}$ und $i,j\geq 1$}} + child[description, level distance=3cm]{node{goto l mit $0\leq l\leq m$ (unbedingter Sprung)}} + child[description, level distance=4.5cm]{node{if $x_i=0$ then l mit $i\geq 1$ und $0\leq l \leq m$ (bedingter Sprung)}} + } + } + child{node [subtopic]{Turing berechenbar} + child[theme, level distance=1cm]{node{Berechnung} + child[description, level distance=1cm]{node{Eingabe $x$ ist anfänglicher Bandinhalt}} + child[description, level distance=2.2cm]{node{Ausgabe $f(x)$ ist Bandinhalt (ohne Blanks am Rand)beim Erreichen eines Endzustands}} + child[description, level distance=3.6cm]{node{Hält die TM für die gegebene Eingabe nicht an, ist $f(x)$ an dieser Stelle undefiniert.}} + } + child[theme, level distance=5.8cm]{node{Church-Turing-These} + child[description, level distance=1.2cm]{node{Alle im intuitiven Sinne berechenbaren Funktionen können mit Turingmaschinen berechnet werden.}} + } }; \end{tikzpicture} -Definition: Sei L eine Sprache. Dann ist $L*=\bigcup_{n\geq 0} L^n$ der Kleene-Abschluss oder die Kleene-Iteration von L. Weiter ist $L+ = \bigcup_{n\geq 0} L^n$ - -\begin{tikzpicture} +\begin{tikzpicture}[ + topic/.style={ + text centered, + text width=6cm, + level distance=1mm, + sibling distance=5mm, + rounded corners=2pt + }, + subtopic/.style={ + yshift=1.5cm, + text centered, + text width=5cm, + rounded corners=2pt, + fill=gray!10 + }, + theme/.style={ + grow=down, + xshift=-1.6cm, + text centered, + text width=5cm, + edge from parent path={(\tikzparentnode.192) |- (\tikzchildnode.west)} + }, + description/.style={ + grow=down, + xshift=-2cm, + text width=5.3cm, + right, + text centered, + edge from parent path={(\tikzparentnode.189) |- (\tikzchildnode.west)} + }, + level 1/.style={sibling distance=7cm}, + level 1/.append style={level distance=2.5cm}, + ] \node[topic]{Rechtslineare Sprachen} child{node [subtopic] {endliche Automaten (Maschinen)} - child[theme, level distance=1cm]{node{deterministischer endlicher Automat M} - child[description, level distance=1cm]{node{5-Tupel $M=(Z, \sum, z_0, \delta, E)$}} - child[description, level distance=1cm]{node{$Z$ eine endliche Menge von Zuständen}} - child[description, level distance=1cm]{node{$\sum$ das Eingabealphabet (mit $Z\cap\sum = \emptyset$)}} - child[description, level distance=1cm]{node{$z_0\in Z$ der Startzustand}} - child[description, level distance=1cm]{node{$\delta: Z \times \sum \rightarrow Z$ die Übergangsfunktion}} - child[description, level distance=1cm]{node{$E\subseteq Z$ die Menge der Endzustände}} - child[description, level distance=1cm]{node{kurz: DFA (deterministic finite automaton)}} - child[description, level distance=1cm]{node{von einem DFA akzeptierte Sprache ist: $L(M)={w\in\sum^* | \hat{\delta}(z_0,w)\in E}$}} - child[description, level distance=1cm]{node{Eine Sprache $L \supseteq \sum^*$ ist regulär, wenn es einen DFA mit $L(M)=L$ gibt}} - %Jede reguläre Sprache ist rechtslinear + child[theme, level distance=1cm]{node{DFA $M$} + child[description, level distance=1.2cm]{node{von einem DFA akzeptierte Sprache ist: $L(M)={w\in\sum^* | \hat{\delta}(z_0,w)\in E}$}} + child[description, level distance=2.6cm]{node{Eine Sprache $L \supseteq \sum^*$ ist regulär, wenn es einen DFA mit $L(M)=L$ gibt}} } - child[theme, level distance=1cm]{node{nicht-deterministischer endlicher Automat M} - %Jede von einem NFA akzeptierte Sprache ist regulär - child[description, level distance=1cm]{node{kurz NFA}} + child[theme, level distance=4.8cm]{node{NFA $M$} + child[description, level distance=1cm]{node {Jede von einem NFA akzeptierte Sprache ist regulär}} } - %Satz: Wenn $L_1$ und $L_2$ reguläre Sprachen sind, dann ist auch $L_1 \cup L_2$ regulär. - %Satz: Wenn $L_1$ und $L_2$ reguläre Sprachen sind, dann ist auch $L_1 \cap L_2$ regulär. - %Satz: Wenn $L_1$ und $L_2$ reguläre Sprachen sind, dann ist auch $L_1L_2$ regulär - %Satz: Wenn L eine reguläre Sprache ist, dann ist auch $L^+/L^*$ regulär + child[theme, level distance=7cm]{node{Satz} + child[description, level distance=1cm]{node {Wenn $L_1$ und $L_2$ reguläre Sprachen sind, dann ist auch}} + child[description, level distance=2cm]{node { $L_1 \cup L_2$ regulär}} + child[description, level distance=3cm]{node {$L_1 \cap L_2$ regulär}} + child[description, level distance=4cm]{node {$L_1L_2$ regulär}} + child[description, level distance=5cm]{node {$L_1^+/L_1^*$ regulär}} + } + child[theme, level distance=13cm]{node{ Jede reguläre Sprache ist rechtslinear}} } - child{description, level distance}{node{Reguläre Ausdrücke - % Definition: Die Menge $Reg(\sum)$ der **regulären Ausdrücke über dem Alphabet $\sum$** ist die kleinste Menge mit folgenden Eigenschaften: - % - $\varnothing \in Reg(\sum), \lambda \in Reg(\sum), \sum \subseteq Reg(\sum)$ - % - Wenn $\alpha, \beta \in Reg(\sum)$, dann auch $(\alpha * \beta), (\alpha + \beta), (\alpha^*)\in Reg(\sum)$ - %- für $\alpha * \beta$ schreibt man oft $\alpha\beta$ - % für $\alpha + \beta$ schreibt man auch $\alpha|\beta$ - - %Für einen regulären Ausdruck $\alpha \in Reg(\sum)$ ist die Sprache $L(\alpha)\subseteq \sum^*$ induktiv definiert - - %zu jedem regulären Ausdruck $\gamma$ gibt es einen NFA M mit $L(\gamma)=L(M)$ - %zu jedem DFA M gibt es einen regulären Ausdruck $\gamma$ mit $L(M)=L(\gamma)$ - }} - %- Rechtslineare Grammatiken - % - Verbindung zur Chomsky Hierarchie - % - erzeugen Sprachen - % - nicht geeignet, um zu entscheiden, ob ein gegebenes Wort zur Sprache gehört - %- NFA - % - erlauben kleine Kompakte Darstellung - % - intuitive graphische Notation - % - nicht geeignet, um zu entscheiden, ob ein gegebenes Wort zur Sprache gehört - %- DFA - % - für effiziente Beantwortung der Frage, ob ein Wort zur Sprache gehört - % - sind uU exponentiell größer als NFA - %- Reguläre Ausdrücke - % - erlauben kompakte Darstellung in Textform - child{node [subtopic] {Nicht-Reguläre Sprachen} - % Für jedes Alphabet $\sum$ existiert eine Sprache L über $\sum$, die von keiner Grammatik G erzeugt wird. - child[theme, level distance=2cm]{node{ Pumping Lemma} - %Wenn L eine reguläre Sprache ist, dann gibt es $n\leq 1$ derart, dass für alle $x\in L$ mit $|x|\geq n$ gilt: es gibt Wörter $u,v,w \in \sum^*$ mit: - %1. $x=uvw$ - %2. $|uv|\leq n$ - %3. $|v|\geq 1$ - %4. $uv^i w\in L$ für alle $i\geq 0$ - - %Dieses Lemma spricht nicht über Automaten, sondern nur über die Eigenschaften der Sprache. Es ist geeignet, Aussagen über Nicht-Regularität zu machen. Dabei ist es aber nur eine notwendige Bedingung. Es kann nicht genutzt werden, um die Regularität einer Sprache L zu zeigen. + child{node [subtopic] {Reguläre Ausdrücke} + child[theme, level distance=1cm]{node {Definition} + child[description, level distance=1.4cm]{node {Die Menge $Reg(\sum)$ der regulären Ausdrücke über dem Alphabet $\sum$ ist die kleinste Menge mit folgenden Eigenschaften:}} + child[description, level distance=2.9cm]{node {$\varnothing \in Reg(\sum), \lambda \in Reg(\sum), \sum \subseteq Reg(\sum)$}} + child[description, level distance=4cm]{node {Wenn $\alpha, \beta \in Reg(\sum)$, dann auch $(\alpha * \beta), (\alpha + \beta), (\alpha^*)\in Reg(\sum)$}} + child[description, level distance=5cm]{node {für $\alpha * \beta$ schreibt man oft $\alpha\beta$}} + child[description, level distance=6cm]{node {für $\alpha + \beta$ schreibt man auch $\alpha|\beta$}} + } + child[theme, level distance=8.5cm]{node {Für einen regulären Ausdruck $\alpha \in Reg(\sum)$ ist die Sprache $L(\alpha)\subseteq \sum^*$ induktiv definiert} + child[description, level distance=1.5cm]{node {zu jedem regulären Ausdruck $\gamma$ gibt es einen NFA M mit $L(\gamma)=L(M)$}} + child[description, level distance=3cm]{node {zu jedem DFA M gibt es einen regulären Ausdruck $\gamma$ mit $L(M)=L(\gamma)$}} + } + } + child{node [subtopic] {Nicht-Reguläre Sprachen} + child[theme, level distance=1cm]{node{ Pumping Lemma} + child[description, level distance=1.5cm]{node {L sei reguläre Sprache, dann gibt es $n\leq 1$ derart, dass für alle $x\in L$ mit $|x|\geq n$ gilt: es gibt Wörter $u,v,w \in \sum^*$ mit}} + child[description, level distance=3cm]{node {$x=uvw$, $|uv|\leq n$, $|v|\geq 1$}} + child[description, level distance=4cm]{node {$uv^i w\in L$ für alle $i\geq 0$}} + child[description, level distance=5cm]{node {geeignet um Aussagen über Nicht-Regularität zu machen}} + } + child[theme, level distance=7cm]{node{ Myhill-Nerode Äquivalenz} + child[description, level distance=1cm]{node {binäre Relation $R_L \subseteq \sum^* \times \sum^*$}} + child[description, level distance=2.3cm]{node {$\forall x,y\in \sum^*$ setze $(x,y)\in R_L$ genau dann, wenn $\forall z \in \sum^* :(xy\in L \leftrightarrow yz \in L)$ gilt. $x R_L y$}} + child[description, level distance=4cm]{node {Für Sprache L und Wort $x\in \sum^*$ ist $[x]_L=\{y\in\sum^* | x R_L y \}$ die Äquivalenzklasse von x}} + child[description, level distance=5.5cm]{node {Satz: L ist regulär $\leftrightarrow index(R_L)< \infty$}} } - child[theme, level distance=3cm]{node{ Myhill-Nerode Äquivalenz} - %Für eine Sprache $L\subseteq \sum^*$ definieren wir eine binäre Relation $R_L \subseteq \sum^* \times \sum^*$ wie folgt: Für alle $x,y\in \sum^*$ setze $(x,y)\in R_L$ genau dann, wenn $\forall z \in \sum^* :(xy\in L \leftrightarrow yz \in L)$ gilt. Wir schreiben hierfür auch $x R_L y$. - % Definition: Für eine Sprache L und ein Wort $x\in \sum^*$ ist $[x]_L=\{y\in\sum^* | x R_L y \}$ die Äquivalenzklasse von x. Ist L klar, so schreiben wir einfacher $[x]$. - %Satz von Myhill-Nerode: Sei L eine Sprache. L ist regulär $\leftrightarrow index(R_L)< \infty$ (d.h. nur wenn die Myhill-Nerode-Äquivalenz endliche Klassen hat) - } } - child{node [subtopic] {Minimalautomat} - %Ein DFA M heißt reduziert, wenn es für jeden Zustand $z \in Z$ ein Wort $x_z\in \sum^*$ gibt mit $\hat{\sigma}(l, x_z)=z$ - } child{node [subtopic] {Entscheidbarkeit} - child[theme, level distance=1cm]{node{Wortproblem}} - child[theme, level distance=1cm]{node{Leerheitsproblem}} - child[theme, level distance=1cm]{node{Endlichkeitsproblem}} - child[theme, level distance=1cm]{node{Schnittproblem}} - child[theme, level distance=1cm]{node{Inklusionsproblem}} - child[theme, level distance=1cm]{node{Äquivalenzproblem}} + child[theme, level distance=1cm]{node{Wortproblem} + child[description, level distance=1cm]{node {Gilt $w\in L$ für eine gegebene reguläre Sprache L und $w\in\sum^*$?}} + } + child[theme, level distance=3cm]{node{Leerheitsproblem} + child[description, level distance=1cm]{node {Gilt $L=\varnothing$ für eine gegebene reguläre Sprache L?}} + } + child[theme, level distance=5cm]{node{Endlichkeitsproblem} + child[description, level distance=1cm]{node {Ist eine gegebene reguläre Sprache L endlich?}} + } + child[theme, level distance=7cm]{node{Schnittproblem} + child[description, level distance=1cm]{node {Gilt $L_1\cap L_2=\varnothing$ für gegebene reguläre $L_1,L_2$?}} + } + child[theme, level distance=9cm]{node{Inklusionsproblem} + child[description, level distance=1cm]{node {Gilt $L_1 \subseteq L_2$ für gegebene reguläre $L_1,L_2$?}} + } + child[theme, level distance=11cm]{node{Äquivalenzproblem} + child[description, level distance=1cm]{node {Gilt $L_1=L_2$ für gegebene reguläre $L_1,L_2$?}} + } }; \end{tikzpicture} -\begin{tikzpicture} - \node[topic]{Kontextfreie Sprachen} - child{node [subtopic] { Ableitungsbäume}} - child{node [subtopic] {Linksableitung}} - child{node [subtopic] {Chomsky Normalform}} - child{node [subtopic] {Der Cocke-Younger-Kasami- oder CYK-Algorithmus}} - child{node [subtopic] {Kellerautomaten}} - child{node [subtopic] {die Greibach-Normalform}} - child{node [subtopic] {PDAs mit Endzuständen}} - child{node [subtopic] {Deterministisch kontextfreie Sprachen}} - child{node [subtopic] {das Pumping Lemma für kontextfreie Sprachen}} - child{node [subtopic] {das Lemma von Ogden (William Ogden)}} - ; -\end{tikzpicture} - -\begin{tikzpicture} - \node[topic]{Berechenbarkeit} - child{node [subtopic] {Loop-Berechenbarkeit}} - child{node [subtopic] {While Programme} - child[theme, level distance=1cm]{node{Gödels Vermutung}} - } - child{node [subtopic] {GoTo Programme} - child[theme, level distance=1cm]{node{Kleenesche Normalform}} - } - child{node [subtopic] {Turing Berechenbarkeit}} - ; -\end{tikzpicture} - -\begin{tikzpicture} - \node[topic]{Entscheidbarkeit} - child{node [subtopic] {Halteproble}} - child{node [subtopic] {Reduktion}} - child{node [subtopic] {Rechnen mit Kodierungen}} - child{node [subtopic] {Satz von Rice}} - child{node [subtopic] {Semi Entscheidbarkeit}} - child{node [subtopic] {Universelle Turing Maschine}} - child{node [subtopic] {Totale berechenbare Funktionen}} - child{node [subtopic] {Einige unentscheidbare Probleme}} - ; -\end{tikzpicture} - -\begin{tikzpicture} +\begin{tikzpicture}[ + topic/.style={ + text centered, + text width=6cm, + level distance=1mm, + sibling distance=5mm, + rounded corners=2pt + }, + subtopic/.style={ + yshift=1.5cm, + text centered, + text width=3.5cm, + rounded corners=2pt, + fill=gray!10 + }, + theme/.style={ + grow=down, + xshift=-1.6cm, + text centered, + text width=4cm, + edge from parent path={(\tikzparentnode.192) |- (\tikzchildnode.west)} + }, + description/.style={ + grow=down, + xshift=-2cm, + text width=3.5cm, + right, + text centered, + edge from parent path={(\tikzparentnode.189) |- (\tikzchildnode.west)} + }, + level 1/.style={sibling distance=4.5cm}, + level 1/.append style={level distance=2.5cm}, + ] \node[topic]{Komplexitätstheorie} child{node [subtopic] {Berechenbarkeitstheorie}} child{node [subtopic] {Frage der Komplexitätstheorie}} child{node [subtopic] {Komplexitätsklassen} child[theme, level distance=1cm]{node{Deterministische Zeitklassen}} - child[theme, level distance=1cm]{node{Deterministische Platzklassen}} - child[theme, level distance=1cm]{node{Nichtdeterministische Zeitklassen}} - child[theme, level distance=1cm]{node{Nichtdeterministische Platzklassen}} + child[theme, level distance=2cm]{node{Deterministische Platzklassen}} + child[theme, level distance=3cm]{node{Nichtdeterministische Zeitklassen}} + child[theme, level distance=4cm]{node{Nichtdeterministische Platzklassen}} } child{node [subtopic] {Polynomialzeit-Reduktionen}} child{node [subtopic] {NP-Vollständigkeit}} child{node [subtopic] {Weitere NP-vollständige Probleme} child[theme, level distance=1cm]{node{3-SAT ist NP-vollständig}} - child[theme, level distance=1cm]{node{3C ist NP-vollständig}} - child[theme, level distance=1cm]{node{DHC ist NP-vollständig}} - child[theme, level distance=1cm]{node{HC ist NP-vollständig}} - child[theme, level distance=1cm]{node{TSP ist NP-vollständige}} + child[theme, level distance=2cm]{node{3C ist NP-vollständig}} + child[theme, level distance=3cm]{node{DHC ist NP-vollständig}} + child[theme, level distance=4cm]{node{HC ist NP-vollständig}} + child[theme, level distance=5cm]{node{TSP ist NP-vollständige}} }; \end{tikzpicture} +\begin{multicols*}{3} + \paragraph{deterministischer endlicher Automat M} + \begin{itemize*} + \item 5-Tupel $M=(Z, \sum, z_0, \delta, E)$ + \item $Z$ eine endliche Menge von Zuständen + \item $\sum$ das Eingabealphabet (mit $Z\cap\sum = \emptyset$) + \item $z_0\in Z$ der Startzustand + \item $\delta: Z \times \sum \rightarrow Z$ die Übergangsfunktion + \item $E\subseteq Z$ die Menge der Endzustände + \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 + \begin{itemize*} + \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, + \item $\delta:Z\times\Phi\rightarrow(Z\times\Phi\times\{L,N,R\})$ die Überführungsfunktion + \item $\Box\in\Phi/\sum$ das Leerzeichen oder Blank und + \item $E\subseteq Z$ die Menge der Endzustände ist + \end{itemize*} + + + \section{Linksableitung} + \section{CYK-Algorithmus} + \section{Kellerautomaten} + \section{die Greibach-Normalform} + \section{das Lemma von Ogden (William Ogden)} + \section{Halteproblem} + \section{Reduktion} + \section{Satz von Rice} + \section{Semi Entscheidbarkeit} + \section{Universelle Turing Maschine} + \section{Totale berechenbare Funktionen} + \section{Einige unentscheidbare Probleme} + +\end{multicols*} + \end{document} diff --git a/Automaten, Sprachen und Komplexität.md b/Automaten, Sprachen und Komplexität.md index 64f7ed1..3972641 100644 --- a/Automaten, Sprachen und Komplexität.md +++ b/Automaten, Sprachen und Komplexität.md @@ -501,7 +501,7 @@ Um zu zeigen, dass eine konkrete Sprache L regulär ist, kann man - zeigen, dass $L=L_1 \cap L_2$ ist und $L_1$ und $L_2$ regulär sind, oder - ... -### Pumping Lemma (auswendig lernen!) +### Pumping Lemma Wenn L eine reguläre Sprache ist, dann gibt es $n\leq 1$ derart, dass für alle $x\in L$ mit $|x|\geq n$ gilt: es gibt Wörter $u,v,w \in \sum^*$ mit: 1. $x=uvw$ 2. $|uv|\leq n$ @@ -604,7 +604,7 @@ Ausgabe: Menge der Paare erkennungsäquivalenter Zustände Fragestellungen/Probleme für reguläre Sprachen ### Wortproblem -Gilt $w\in L$ für eine gegebene reguläre Sprache L und $w\in\sum^*$ +Gilt $w\in L$ für eine gegebene reguläre Sprache L und $w\in\sum^*$? Eingabe: DFA M und $w\in\sum^*$ @@ -1225,7 +1225,7 @@ Loop berechenbare Funktionen können sehr schnell wachsen, die Ackermann Funktio Konstruktion: Für $f:\N\rightarrow\N$ sei $F(f)=g:\N\rightarrow\N$ definiert durch $$g(y)=\begin{cases} f(1)\quad\text{falls } y=0\\ f(g(y-1)) \quad\text{falls } y>0\end{cases}$$ Also ist $F:\N^{\N}\rightarrow\N^{\N}$ Funktion, die numerische Funktionen auf numerische Funktionen abbildet. Wir definieren nun ein Folge von Funktionen $ack_x:\N\rightarrow\N$ für $x\in\N$: - $ack_0:\N\rightarrow\N:y\rightarrow y+1$ -- $ack_{x+1}=F(ack_x), d.h. +- $ack_{x+1}=F(ack_x)$, d.h. - $ack_{x+1}(y) = \begin{cases} ack_x(1) \quad\text{falls } y=0\\ ack_x(ack_{x+1}(y-1)) \quad\text{falls } y>0 \end{cases}$ > Definition: Die Funktion $ack:\N^2\rightarrow\N$ mit $ack(x,y,)=ack_x(y)$ heißt Ackermann Funktion @@ -1344,7 +1344,7 @@ Idee: - Dann wieder nach links laufen und jede 1 durch 0 ersetzen, solange bis eine 0 oder ein Leerzeichen auftaucht. - Dieses Zeichen dann durch 1 ersetzen, bis zum Zahlanfang laufen und in einen Endzustand übergehen. -> Definition: Eine Turingmaschine (TM) ist ein 7-Tupel $M=(Z,\sum, \Phi, \delta, z_o, \Box, E)$, weobei +> Definition: Eine Turingmaschine (TM) ist ein 7-Tupel $M=(Z,\sum, \Phi, \delta, z_o, \Box, E)$, wobei > - $\sum$ das Eingabealphabet > - $\Phi$ mit $\Phi\supseteq\sum$ und $\Phi\cap Z\not= 0$ das Arbeits- oder Bandalphabet, > - $z_0\in Z$ der Startzustand,