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+\documentclass[a4paper]{article} +\usepackage[ngerman]{babel} +\usepackage{multicol} +\usepackage{calc} +\usepackage{ifthen} +\usepackage[landscape,left=1cm,top=1cm,right=1cm,nohead,nofoot]{geometry} +\usepackage{amsmath,amsthm,amsfonts,amssymb} +\usepackage{color,graphicx,overpic} +\usepackage{listings} +\usepackage[compact]{titlesec} %less space for headers +\usepackage{mdwlist} %less space for lists \usepackage[utf8]{inputenc} \usepackage{tikz} -\usetikzlibrary{mindmap} +\usepackage{pdflscape} +\usepackage{verbatim} +\usetikzlibrary{mindmap, arrows,shapes,positioning,shadows,trees} +\tikzstyle{every node}=[draw=black,thin,anchor=west, minimum height=2em] +\usepackage[hidelinks,pdfencoding=auto]{hyperref} +\pdfinfo{ + /Title (Automaten, Sprachen \& Komplexität - Cheatsheet) + /Creator (TeX) + /Producer (pdfTeX 1.40.0) + /Author (Robert Jeutter) + /Subject () +} +% Information boxes +\newcommand*{\info}[4][16.3]{ + \node [ annotation, #3, scale=0.65, text width = #1em, inner sep = 2mm ] at (#2) { + \list{$\bullet$}{\topsep=0pt\itemsep=0pt\parsep=0pt + \parskip=0pt\labelwidth=8pt\leftmargin=8pt + \itemindent=0pt\labelsep=2pt} + #4 + \endlist + }; +} + +% This sets page margins to .5 inch if using letter paper, and to 1cm +% if using A4 paper. (This probably isn't strictly necessary.) +% If using another size paper, use default 1cm margins. +\ifthenelse{\lengthtest { \paperwidth = 11in}} + { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } + {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} + {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } + {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } + } + +% Redefine section commands to use less space +\makeatletter +\renewcommand{\section}{\@startsection{section}{1}{0mm}% + {-1ex plus -.5ex minus -.2ex}% + {0.5ex plus .2ex}%x + {\normalfont\large\bfseries}} +\renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% + {-1explus -.5ex minus -.2ex}% + {0.5ex plus .2ex}% + {\normalfont\normalsize\bfseries}} +\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% + {-1ex plus -.5ex minus -.2ex}% + {1ex plus .2ex}% + {\normalfont\small\bfseries}} +\makeatother + +% Define BibTeX command +\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em + T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} + +% Don't print section numbers +\setcounter{secnumdepth}{0} + +\setlength{\parindent}{0pt} +\setlength{\parskip}{0pt plus 0.5ex} +% compress space +\setlength\abovedisplayskip{0pt} +\setlength{\parskip}{0pt} +\setlength{\parsep}{0pt} +\setlength{\topskip}{0pt} +\setlength{\topsep}{0pt} +\setlength{\partopsep}{0pt} +\linespread{0.5} +\titlespacing{\section}{0pt}{*0}{*0} +\titlespacing{\subsection}{0pt}{*0}{*0} +\titlespacing{\subsubsection}{0pt}{*0}{*0} + +%My Environments +\newtheorem{example}[section]{Example} + +%Tikz global setting +\tikzset{ + topic/.style={ + text centered, + text width=5cm, + level distance=1mm, + sibling distance=5mm, + rounded corners=2pt + }, + subtopic/.style={ + yshift=1.5cm, + text centered, + text width=3cm, + rounded corners=2pt, + fill=gray!10 + }, + theme/.style={ + grow=down, + xshift=-0.6cm, + text centered, + text width=3cm, + edge from parent path={(\tikzparentnode.205) |- (\tikzchildnode.west)} + }, + description/.style={ + grow=down, + xshift=-0.5cm, + right, + text centered, + edge from parent path={(\tikzparentnode.200) |- (\tikzchildnode.west)} + }, + level1/.style ={level distance=1cm}, + level2/.style ={level distance=2cm}, + level3/.style ={level distance=3cm}, + level4/.style ={level distance=4cm}, + level5/.style ={level distance=5cm}, + level6/.style ={level distance=6cm}, + level7/.style ={level distance=7cm}, + level8/.style ={level distance=8cm}, + level9/.style ={level distance=9cm}, + level 1/.style={sibling distance=5.5cm}, + level 1/.append style={level distance=2.5cm}, +} + +% Turn off header and footer \pagestyle{empty} \begin{document} -\begin{tikzpicture}[mindmap, grow cyclic, every node/.style=concept, concept color=orange!40, - level 1/.append style={level distance=5cm,sibling angle=90}, - level 2/.append style={level distance=3cm,sibling angle=45}] - -\node{Sprache} - child [concept color=blue!30] { node {Chomsky Hierachie} - child { node {Typ 0: Allgemein}} - child { node {Typ 1: Kontextsensitiv}} - child { node {Typ 2: Kontextfrei}} - child { node {Typ 3: Regulär}} - } - child [concept color=yellow!30] { node {Wort} - child { node { y Präfix von w, wenn es $z\in\sum^*$ gibt mit $yz=w$}} - child { node { y Infix/Faktor von w, wenn es $x,z\in\sum^*$ gibt mit $xyz = w$}} - child { node { y Suffix von w, wenn es $x\in\sum^*$ gibt mit $xy=w$}} - } - child [concept color=teal!40] { node {Beamer Series} - } - child [concept color=purple!50] { node {TikZ Series} - }; +\begin{tikzpicture} + \node[topic]{Automaten, Sprachen \& Komplexität} + child{node [subtopic]{Sprache} + child [theme, level1] { node {Chomsky Hierachie} + child[description, level distance=1cm] { node {Typ 0: Allgemein \\ jede Grammatik ist vom Typ 0}} + child[description, level distance=2cm] { 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=3cm] { node {Typ 2: Kontextfrei \\ wenn $l\in V$ und $r\in (V\cup \sum)^*$ gilt}} + child[description, level distance=4cm] { node {Typ 3: Regulär \\ wenn $l\in V$ und $r\in \sum V\cup {\epsilon}$ gilt}} + } + } + 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]{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[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=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}[mindmap, grow cyclic, every node/.style=concept, concept color=orange!40, - level 1/.append style={level distance=5cm,sibling angle=90}, - level 2/.append style={level distance=3cm,sibling angle=45}] -\node{Grammatiken} - child [concept color=blue!30] { node {Symbole} - child { node {Nicht-Terminale (oder Variablen), aus denen noch weitere Wortbestandteile abgeleitet werden sollen}} - child { node {Terminale (die "eigentlichen" Symbole)}} +\begin{tikzpicture}[ + subtopic/.style={ + yshift=1.5cm, + text centered, + text width=3cm, + rounded corners=2pt, + fill=gray!10 + }, + level 1/.style={sibling distance=5.5cm}, + 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}} + }; +\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} + \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 [concept color=yellow!30] { node {4-Tupel $G=(V,\sum, P, S)$} - child { node {V ist eine endliche Menge von Nicht-Terminalen oder Variablen (Bsp $V=\{S,B,C\}$)}} - child { node {$\sum$ ist ein Alphabet (Menge der Terminale) mit $V\cap \sum= \varnothing$, d.h. kein Zeichen ist gleichzeitig Terminal und Nicht-Terminal (Bsp $\sum=\{a,b,c\}$)}} - child { node {$P\subseteq (V\cup \sum)^+ \times (v\cup\sum)^*$ ist eine endliche Menge von Regeln oder Produktionen (Produktionsmenge) (Bsp $P=\{(S,aSBC), (S,aBC), (cC,cc), (aB,ab)\}$)}} - child { node {$S\in V$ ist das Startsymbol/ die Startvariable oder das Axiom}} + 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}} + } + %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{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[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}} + }; +\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} + \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{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}} + }; +\end{tikzpicture} + \end{document} diff --git a/Automaten, Sprachen und Komplexität.md b/Automaten, Sprachen und Komplexität.md index 3d37d46..64f7ed1 100644 --- a/Automaten, Sprachen und Komplexität.md +++ b/Automaten, Sprachen und Komplexität.md @@ -4,6 +4,108 @@ date: Wintersemester 20/21 author: Robert Jeutter --- +- [Einführung](#einführung) + - [Grundfrage](#grundfrage) + - [Probleme (als Abbildung)](#probleme-als-abbildung) + - [(beschränkte) Resourcen](#beschränkte-resourcen) +- [Grundbegriffe](#grundbegriffe) + - [Chomsky Hierarchie](#chomsky-hierarchie) +- [Rechtslineare Sprachen](#rechtslineare-sprachen) + - [endliche Automaten (Maschinen)](#endliche-automaten-maschinen) + - [Reguläre Ausdrücke](#reguläre-ausdrücke) + - [Zusammenfassung](#zusammenfassung) + - [Nicht-Reguläre Sprachen](#nicht-reguläre-sprachen) + - [Konkrete nicht-reguläre Sprachen](#konkrete-nicht-reguläre-sprachen) + - [Pumping Lemma (auswendig lernen!)](#pumping-lemma-auswendig-lernen) + - [Myhill-Nerode Äquivalenz](#myhill-nerode-äquivalenz) + - [Minimalautomat](#minimalautomat) + - [Algorithmus Minimalautomat](#algorithmus-minimalautomat) + - [Entscheidbarkeit](#entscheidbarkeit) + - [Wortproblem](#wortproblem) + - [Leerheitsproblem](#leerheitsproblem) + - [Endlichkeitsproblem](#endlichkeitsproblem) + - [Schnittproblem](#schnittproblem) + - [Inklusionsproblem](#inklusionsproblem) + - [Äquivalenzproblem](#äquivalenzproblem) + - [Effizientbetrachtung](#effizientbetrachtung) + - [Pumping Lemma mit Alphabet aus einem Zeichen](#pumping-lemma-mit-alphabet-aus-einem-zeichen) + - [Spielschema oder anderes Schema in Prüfung gefirdert](#spielschema-oder-anderes-schema-in-prüfung-gefirdert) + - [Produktbildung von zwei regulären Sprachen. Wenn die erste Sprache als Startzustand da leere Wort enthält, muss man den Startzustand der zweiten Sprache beibehalten?](#produktbildung-von-zwei-regulären-sprachen-wenn-die-erste-sprache-als-startzustand-da-leere-wort-enthält-muss-man-den-startzustand-der-zweiten-sprache-beibehalten) +- [Kontextfreie Sprachen](#kontextfreie-sprachen) + - [Ableitungsbäume](#ableitungsbäume) + - [Linksableitung](#linksableitung) + - [kontextfreie Sprachen sind kontext-sensitiv](#kontextfreie-sprachen-sind-kontext-sensitiv) + - [Chomsky Normalform](#chomsky-normalform) + - [Der Cocke-Younger-Kasami- oder CYK-Algorithmus](#der-cocke-younger-kasami--oder-cyk-algorithmus) + - [Kellerautomaten](#kellerautomaten) + - [die Greibach-Normalform](#die-greibach-normalform) + - [Von Grammatiken zu PDAs](#von-grammatiken-zu-pdas) + - [Von PDAs zu Grammatiken](#von-pdas-zu-grammatiken) + - [PDAs mit Endzuständen](#pdas-mit-endzuständen) + - [Deterministisch kontextfreie Sprachen](#deterministisch-kontextfreie-sprachen) + - [Abschlusseigenschaften](#abschlusseigenschaften) + - [das Pumping Lemma für kontextfreie Sprachen](#das-pumping-lemma-für-kontextfreie-sprachen) + - [das Lemma von Ogden (William Ogden)](#das-lemma-von-ogden-william-ogden) + - [Entscheidbarkeit](#entscheidbarkeit-1) + - [2. dann zeige $W\supseteq W_{|V|}$](#2-dann-zeige-wsupseteq-w_v) + - [Unentscheidbarkeit bei kontextfreien Sprachen](#unentscheidbarkeit-bei-kontextfreien-sprachen) + - [Entscheidbarkeit bei deterministisch kontextfreien Sprachen](#entscheidbarkeit-bei-deterministisch-kontextfreien-sprachen) + - [Unentscheidbarkeit bei deterministisch kontextfreien Sprachen](#unentscheidbarkeit-bei-deterministisch-kontextfreien-sprachen) + - [Zusammenfassung kontextfreie Sprachen](#zusammenfassung-kontextfreie-sprachen) +- [Berechenbarkeit](#berechenbarkeit) + - [Loop-Berechenbarkeit](#loop-berechenbarkeit) + - [(K+) viele Loop-berechenbare Funktionen](#k-viele-loop-berechenbare-funktionen) + - [(A+) viele Abschlusseigenschaften](#a-viele-abschlusseigenschaften) + - [Primitiv-rekursive Funktionen](#primitiv-rekursive-funktionen) + - [Argument K- gegen die Loop Vermutung](#argument-k--gegen-die-loop-vermutung) + - [Ackermann Funktion](#ackermann-funktion) + - [While Programme](#while-programme) + - [Gödels Vermutung](#gödels-vermutung) + - [GoTo Programme](#goto-programme) + - [Ein kleiner Ausflug - Kleenesche Normalform](#ein-kleiner-ausflug---kleenesche-normalform) + - [Turing Berechenbarkeit](#turing-berechenbarkeit) + - [Beispiel einer Turingmaschine (intuitiv)](#beispiel-einer-turingmaschine-intuitiv) + - [Beispiel Turingmaschine (formal)](#beispiel-turingmaschine-formal) + - [Mehrband Tunringmaschine](#mehrband-tunringmaschine) + - [Ausflug: Zählermaschine](#ausflug-zählermaschine) +- [Entscheidbarkeit](#entscheidbarkeit-2) + - [Halteproble](#halteproble) + - [Reduktion](#reduktion) + - [Rechnen mit Kodierungen](#rechnen-mit-kodierungen) + - [Satz von Rice](#satz-von-rice) + - [Semi Entscheidbarkeit](#semi-entscheidbarkeit) + - [Universelle Turing Maschine](#universelle-turing-maschine) + - [Totale berechenbare Funktionen](#totale-berechenbare-funktionen) + - [Einige unentscheidbare Probleme](#einige-unentscheidbare-probleme) + - [Kontextfreie Sprachen](#kontextfreie-sprachen-1) +- [Komplexitätstheorie](#komplexitätstheorie) + - [Zusammenfassung Berechenbarkeitstheorie](#zusammenfassung-berechenbarkeitstheorie) + - [Die zentrale Frage der Komplexitätstheorie](#die-zentrale-frage-der-komplexitätstheorie) + - [Komplexitätsklassen](#komplexitätsklassen) + - [Deterministische Zeitklassen](#deterministische-zeitklassen) + - [Einige typische Probleme in P](#einige-typische-probleme-in-p) + - [Erreichbarkeit](#erreichbarkeit) + - [Euler-Kreise](#euler-kreise) + - [Deterministische Platzklassen](#deterministische-platzklassen) + - [Einige typische Probleme in PSPACE: Erfüllbarkeit](#einige-typische-probleme-in-pspace-erfüllbarkeit) + - [Einige typische Probleme in PSPACE: Hamilton-Kreise](#einige-typische-probleme-in-pspace-hamilton-kreise) + - [Einige typische Probleme in PSPACE: 3-Färbbarkeit](#einige-typische-probleme-in-pspace-3-färbbarkeit) + - [Zusammenfassung: typische Probleme](#zusammenfassung-typische-probleme) + - [Nichtdeterministische Turingmaschinen](#nichtdeterministische-turingmaschinen) + - [Determinisierbarkeit von NTM](#determinisierbarkeit-von-ntm) + - [Nichtdeterministische Zeitklassen](#nichtdeterministische-zeitklassen) + - [Nichtdeterministische Platzklassen](#nichtdeterministische-platzklassen) + - [Typische Probleme, 2. Versuch](#typische-probleme-2-versuch) + - [Polynomialzeit-Reduktionen](#polynomialzeit-reduktionen) + - [NP-Vollständigkeit](#np-vollständigkeit) + - [Weitere NP-vollständige Probleme](#weitere-np-vollständige-probleme) + - [3-SAT ist NP-vollständig](#3-sat-ist-np-vollständig) + - [3C ist NP-vollständig](#3c-ist-np-vollständig) + - [DHC ist NP-vollständig](#dhc-ist-np-vollständig) + - [HC ist NP-vollständig](#hc-ist-np-vollständig) + - [TSP ist NP-vollständig](#tsp-ist-np-vollständig) + - [Zusammenfassung](#zusammenfassung-1) + Literaturempfehlung: Theoretische Informatik - kurz gefasst, Uwe Schöning, Spektrum Akademischer Weg # Einführung @@ -139,7 +241,9 @@ Konventionen: - für $(l,r)\in P$ schreibt man auch $l\rightarrow r$ > Definition: Sei $G=(V, \sum, P, S)$ eine Grammatik. Eine **Ableitung** ist eine endliche Folge von Wörtern +> > Ein **Wort** $w\in (V\cup\sum)^*$ heißt Satzform, wenn es eine Ableitung gibt, deren letztes Wort w ist. +> > Die **Sprache** $L(G)={w\in \sum^* | S\Rightarrow_G^* w}$ aller Satzformen aus $\sum^*$ heißt von G erzeugte Sprache. Dabei ist $\Rightarrow_G^*$ der reflexive und transitive Abschluss von $\Rightarrow_G$. D.h. die von G erzeugte Sprache L(G) besteht genau aus den Wörtern, die in beliebig vielen Schritten aus S abgeleitet werden können und nur aus Terminalen besteht. @@ -154,14 +258,14 @@ Nichtdeterminismus kann verursacht werden durch: - manchmal können Ableitungen in einer Sackgasse enden, d.h. obwohl noch nichtterminale in einer Satzformen vorkommen, ist keine Regel mehr anwendbar. ## Chomsky Hierarchie -. Typ 0 (Chomsky-0): Jede Grammatik ist vom Typ 0 (Semi-Thue-System) +- Typ 0 (Chomsky-0): Jede Grammatik ist vom Typ 0 (Semi-Thue-System) - Typ 1: Eine Regel heißt kontext-sensitiv, wenn es Wörter $u,v,w\in(V\cup\sum)^*,|v|>0$ und ein Nichtterminal $A\in V$ gibt mit $l=uAw$ und $r=uvw$. Eine Grammatik ist vom Typ 1 (oder kontext-sensitiv) falls - alle Regeln aus P kontext-sensitiv sind - $(S\rightarrow \epsilon)\in P$ die einzige nicht kontext-sensitive Regel in P ist und S auf keiner rechten Seite einer Regel aus P vorkommt - Typ 2: eine Regel $(l\rightarrow r)$ heißt kontext-frei wenn $l\in V$ und $r\in (V\cup \sum)^*$ gilt. Eine Grammatik ist vom Typ 2, falls sie nur kontext-freie Regeln enthält - Typ 3: Eine Regl ist rechtslinear, wenn $l\in V$ und $r\in \sum V\cup {\epsilon}$ gilt. Eine Grammatik ist vom Typ 3 wenn sie nur rechtslineare Regeln enthält -> Definition: Eine Sprache heißt vom Typ i (4i\in {0,1,2,3}$) falls es eine Typ-i-Grammatik gibt mit $L(G)=L$. Wir bezeichnen mit $L$, die Klasse der Sprache vom Typ i. +> Definition: Eine Sprache heißt vom Typ i ($i\in \{0,1,2,3\}$) falls es eine Typ-i-Grammatik gibt mit $L(G)=L$. Wir bezeichnen mit $L$, die Klasse der Sprache vom Typ i. Eine Sprache vom Typ i nennt man auch rekursiv aufzählbar (i=0, RE), kontext-sensitiv (i=1, CS), kontext-frei (i=2, CF) oder rechtslinear (i=3, REG). @@ -435,8 +539,8 @@ Der Index $index(R)$ von R ist die Anzahl der Äquivalenzklassen von R: $index(R (d.h. nur wenn die Myhill-Nerode-Äquivalenz endliche Klassen hat) Beweis: -- "$Rightarrow$": Sei L regulär -> es gibt DFA M mit $L(M)=L$... -- "$Leftarrow$": sei $index(R_L)< \infty$ -> Definiere einen DFA $M_L=(\{[x_1],...,[x_n]\},\sum,[\epsilon],\sigma,\{[w]|w\in L\})$ +- "$\Rightarrow$": Sei L regulär -> es gibt DFA M mit $L(M)=L$... +- "$\Leftarrow$": sei $index(R_L)< \infty$ -> Definiere einen DFA $M_L=(\{[x_1],...,[x_n]\},\sum,[\epsilon],\sigma,\{[w]|w\in L\})$ ## Minimalautomat @@ -457,7 +561,8 @@ Wenn in einem DFA M aus Startzustand X und Y dieselben Sprachen akzeptiert werde > Definition: Sei M ein DFA. Dann ist $M'=(Z_{\equiv},\sum, [z_0],\sigma', E')$ mit > - $\sigma'([z],a)=[\sigma (z,a)]$ für $z\in Z$ und $a\in \sum$ und -> - $E'=\{[z]|z\in E\} +> - $E'=\{[z]|z\in E\}$ +> > der Quotient von M bzgl $\equiv$ (es wird nicht mehr jeder einzelne Fall betrachtet sondern "ganze Gruppen"; Bsp Sitz->Reihe) @@ -1985,7 +2090,7 @@ Wir haben gezeigt: Die Grenze zwischen „einfachen“ und „schwierigen“ Formeln liegt also zwischen Formeln in KNF mit höchstens zwei bzw. höchstens drei Literalen pro Klausel. -## 3C ist NP-vollständig +### 3C ist NP-vollständig k-Färbbarkeit von Graphen - EINGABE: Ein ungerichteter Graph $G = (V , E )$. - FRAGE: Gibt es Zuordnung von k verschiedenen Farben zu Knoten in V, so dass keine zwei benachbarten Knoten $v_1,v_2$ dieselbe Farbe haben? @@ -2012,7 +2117,7 @@ Sei also $\phi$ Formel in KNF, deren Klauseln genau drei Literale enthalten (ggf ![Dreieckfärbung](Assets/ASK_Dreieckfaerbung4.png) -## DHC ist NP-vollständig +### DHC ist NP-vollständig > DHC - Gerichteter Hamiltonkreis > - EINGABE: ein gerichteter Graph $G = (V , E )$ mit Knotenmenge $V$ und Kantenmenge $E\supseteq V\times V$. > - FRAGE: Besitzt der Graph G einen Hamiltonkreis, d.h. kann man den Graphen so durchlaufen, dass jeder Knoten genau einmal besucht wird? @@ -2050,7 +2155,7 @@ Damit gilt für den so definierten Graphen G: - G kann aus $\phi$ in polynomieller Zeit berechnet werden. also: $3-SAT \leq_P DHC$, womit folgt, daß DHC NP-hart und damit NP-vollständig ist. -## HC ist NP-vollständig +### HC ist NP-vollständig Im nächsten Schritt zeigen wir, dass auch das analoge Problem für ungerichtete Graphen NP-vollständig ist. > HC - Ungerichteter Hamiltonkreis @@ -2065,7 +2170,7 @@ Im nächsten Schritt zeigen wir, dass auch das analoge Problem für ungerichtete - Idee: Ersetze einen Knoten mit ein- und ausgehenden Kanten wie folgt ![Hamiltonkreiskanten](Assets/ASK_HamiltonkreisKanten.png) -## TSP ist NP-vollständig +### TSP ist NP-vollständig Wir zeigen nun, daß auch das Travelling-Salesman-Problem NP-vollständig ist. > TSP - Travelling Salesman