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Info_3_Skript_WS2016-17.tex
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Info_3_Skript_WS2016-17.tex
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% Vorlesungsskript / -mitschrieb zu Informatik III - Theoretische Informatik, gehalten von Prof. Dr. Peter Thiemann im WS 2014/15
% Copyright (C) 2016 Ralph Lesch
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program. If not, see <http://www.gnu.org/licenses/>.
% Compiled with pdflatex -enable-write18 -synctex=1 -interaction=nonstopmode --shell-escape %.tex
%\RequirePackage[l2tabu,orthodox]{nag}
\documentclass[11pt,paper=a4,titlepage,headsepline,ngerman,listof=totoc]{scrartcl}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{textcomp}
\usepackage{babel}
\usepackage{needspace}
%
\usepackage{acronym}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{array}
\usepackage{booktabs} % Better rules: \toprule, \midrule, \bottomrule
\usepackage{calc}
\usepackage{cancel}
\usepackage[iso]{datetime} % Date format
\usepackage{enumitem}
\usepackage{float} % H - option for figure
\usepackage{graphicx}
\usepackage{listings} % Source printing & syntax highlighting
\usepackage{mathtools}
\usepackage{multirow}
\usepackage{placeins} % \FloatBarrier
\usepackage{stmaryrd} % Symbols like \llbracket for [[
\usepackage{tabu}
\usepackage[symbol,perpage]{footmisc} % custom footnote markers
\DefineFNsymbols{daggers}{{$\dagger$}{$\dagger\dagger$}{${\dagger}{\dagger}{\dagger}$}{${\dagger}{\dagger}{\dagger}{\dagger}$}{${\dagger}{\dagger}{\dagger}{\dagger}{\dagger}$}}
\setfnsymbol{daggers}
% \renewcommand{\thefootnote}{\fnsymbol{footnote}}
% \usepackage[charter]{mathdesign}
% \KOMAoptions{DIV=calc}
% physics package, without redefinitions
\usepackage[log-declarations=false]{xparse} % Fix for invalid error file reference for errors in log file (because of parentheses in d() arguments).
\usepackage[notrig]{physics}
\let\div\divisionsymbol % \div = tex: \div | physics: \divergence
\let\Real\Re \let\Re\real % \Re = tex: \Re | physics: \Re => \Real
\let\Imaginary\Im \let\Im\imaginary % \Im = tex: \Im | physics \Im => \Imaginary
\usepackage[dvipsnames]{xcolor}
% \usepackage{ulem} % emph = \underline
\newcommand\coloruline[2]{\colorlet{colorsave}{.}{\color{#1}\uline{{\color{colorsave}#2}}}}
%\usepackage{xhfill}
% References
\usepackage[bookmarksnumbered,colorlinks=false,linkbordercolor={0 0 0},pdfborder={0 0 0}]{hyperref}
%\usepackage{hypcap}
\usepackage[hypcap]{caption} % \captionof + hypcap = link at begin of figure etc.
\usepackage{subcaption} % subfigure
%\usepackage{lastpage}
% Custom packages and macros
%\usepackage{import} \subimport{..\}{latexmacros.tex}
\input{latexmacros.tex}
% == fixme ==
\usepackage{fixme}
% Register users for user commands
\FXRegisterAuthor{rl}{anrl}{RL} % Ralph Lesch => \rlnote = \txnote[user=RL]
\FXRegisterAuthor{pt}{anpt}{\bfseries PT} % Peter Thiemann
\FXRegisterAuthor{date}{andate}{\color{black}Vorlesung} % \datenote = Lecture date
\FXRegisterAuthor{draft}{andraft}{\color{red}Entwurf für Vorlesung} % \draftnote = Lecture date
%
\fxsetup{
status=draft,
multiuser,
theme=color,
innerlayout={layout=marginnote}, % fxnotes also in displaymath...
% targetface=\color{red},
% marginface=\color{red},
% inlineface=\color{red}
}
\fxloadlayouts{marginnote}
% For color theme/layout:
\definecolor{fxnote}{named}{Green}
\definecolor{fxwarning}{named}{Orange}
\definecolor{fxerror}{named}{red}
\definecolor{fxfatal}{named}{BrickRed}
\renewcommand*{\marginfont}{\color{red}}
%\renewcommand*{\fxnotename}[1]{\!} % Not named for noncolor theme
%\definecolor{fxtarget}{named}{red} % color for target with color layout/theme
\renewcommand\germanlistfixmename{Anmerkungsverzeichnis}
% Customisation
\makeatletter
% target color = note (type) color
\renewcommand\FXTargetLayoutColor[2]{\@fxuseface{target}\color{fx#1}#2}
% Inline: with braces []
\renewcommand*\FXLayoutInline[3]{%
\@fxdocolon{#3}{\@fxuseface{inline}\color{fx#1}[\ignorespaces#3\@fxcolon#2]}%
}
% Color theme: marginnote like marginpar
\renewcommand*\FXLayoutMarginNote[3]{%
\@fxdocolon{#3}%
\marginnote[%
\raggedleft\@fxuseface{margin}\color{fx#1}\ignorespaces#3\@fxcolon#2%
]{%
\raggedright\@fxuseface{margin}\color{fx#1}\ignorespaces#3\@fxcolon#2%
}%
}
\makeatother
% ====
% == tikz ==
\usepackage{tikz}
\usepackage{tikz-qtree}
\usetikzlibrary{arrows.meta,automata,calc,decorations.pathmorphing,decorations.pathreplacing,graphs,positioning,shapes%
%,external
}
%\tikzexternalize
\tikzset{>=stealth,
block/.style={% Nodes as blocks with aligned text, e.g. for Turing machines
draw, rectangle,
%minimum height=1em,
minimum width=1.5em,
outer sep=0pt,
node distance=0pt,
text height=2ex,
text depth=.5ex,
align=center
},
circle/.style={
draw,
shape=circle,
minimum size=0.5cm,
text=black,
text width=0.5cm,
align=center
}
}
% ====
% == Header and footer ==
\usepackage{scrlayer-scrpage}
\automark[subsection]{section}
\pagestyle{scrheadings}
\clearscrheadfoot
\setkomafont{pageheadfoot}{\normalfont\sffamily}
\ohead{\pagemark}
\ihead{\rightmark}
%\ofoot{\usekomafont{pagenumber}\pagemark/\pageref{LastPage}} % Site number
% == Style ==
\setkomafont{captionlabel}{\bfseries}
% Paragraph
\setlength{\parindent}{0pt}
\setlength{\parskip}{\medskipamount}
% == Environments ==
\usepackage{amsthm,thmtools}
% Fix for theoream key "restate" with name={[short name]}
\makeatletter
\kv@set@family@handler{restate phase 2}{%
% \ifthmt@restatethis
% \@xa\@xa\@xa\g@addto@macro\@xa\@xa\@xa\thmt@storedoptargs\@xa\@xa\@xa{\@xa\@xa\@xa,%
% \@xa\kv@key\@xa=\kv@value}%
% \fi
}
\makeatother
% German "continues" text
\renewcommand\thmcontinues[1]{%
\ifcsname hyperref\endcsname%
\hyperref[#1]{Fortsetzung}%
\else%
Fortsetzung%
\fi%
von S.\,\pageref{#1}%
}
% Theorem Styles
\declaretheoremstyle[
headfont=\bfseries,%\scshape
notefont=\normalfont,
bodyfont=\normalfont,
headpunct={:\ },
qed={},
spaceabove=\bigskipamount,
spacebelow=\parskip
]{basic}
\declaretheoremstyle[
headfont=\bfseries,%\scshape
notefont=\normalfont,
bodyfont=\normalfont,
headpunct={:\ },
qed={\ensuremath{\bigoplus}},
spaceabove=\bigskipamount,
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]{definition}
\declaretheoremstyle[
headfont=\scshape,%\bfseries,
notefont=\normalfont,
bodyfont=\normalfont,
headpunct={:\ },
qed=\openbox,
spaceabove=\parskip,
spacebelow=\parskip
]{proofstyle}
%\theoremstyle{theorem}
\declaretheorem[style=basic,numbered=no,name=Bemerkung]{Bemerkung}
\declaretheorem[style=basic,numbered=no,name=Bem.]{Bem}
\declaretheorem[style=basic,numbered=no,name=Beobachtung]{Beobachtung}
\declaretheorem[style=basic,parent=section,name=Bsp.]{Bsp}
\declaretheorem[style=basic,name=Bsp.,numbered=no]{Bsp*}
\declaretheorem[style=definition,parent=section,name=Def.]{Def}
\declaretheorem[style=definition,name=Def.,numbered=no]{Def*}
\declaretheorem[style=definition,parent=Def,name=Def.]{subDef}
\declaretheorem[style=definition,numbered=no,name=Erinnerung]{Erinnerung}
\declaretheorem[style=basic,parent=section,name=Satz]{Satz}
\declaretheorem[style=basic,name=Satz,numbered=no]{Satz*}
\declaretheorem[style=basic,name=These,numbered=no]{These*}
\declaretheorem[style=basic,numbered=no,name=Lemma]{lemma*}
\declaretheorem[style=basic,sibling=Satz,name=Lemma]{lemma}
\declaretheorem[style=definition,sibling=Satz,name=Korollar]{Korollar}
\declaretheorem[style=definition,numbered=no,name=Korollar]{Korollar*}
\let\proof=\relax
\AtBeginDocument{% For \proofname as of babel.
\declaretheorem[style=proofstyle,name=\proofname,numbered=no]{proof}
}
% cases environment with 2 arrows
% Source: https://tex.stackexchange.com/questions/89250/modify-case-equations-brace/89257#89257
% \splitlines[->] for arrow
\newcommand*{\splitlines}[1][]{
\begin{tikzpicture}[baseline=-0.5ex]
\draw[#1] (0,0) -- (0.4,0.25);
\draw[#1] (0,0) -- (0.4,-0.15);
\end{tikzpicture}
}
\newenvironment{casesarrows}[1][->]%
{\;\splitlines[#1]\;\begin{array}{@{}l@{}}}%
{\end{array}}
% == Custom commands ==
% Source: https://tex.stackexchange.com/questions/164506/how-to-get-a-curved-arrow-pointing-left-and-right/164511#164511
%\newcommand{\curvearrowleftright}{\scalebox{1.2}[2]{$\mathclap{\curvearrowleft}\mkern2.2mu\mathclap{\curvearrowright}$}}
\newcommand{\curvearrowleftright}{\mathrel{\curvearrowleft\mkern-2.7mu\mathllap{\curvearrowright}}}
% \dash of given length
\newcommand*{\xdash}[1]{\rule[0.5ex]{#1}{0.55pt}}
% \ruleplaceholder["]{width} = --"--
\newcommand*{\ruleplaceholder}[2][\text{\texttt{ " }}]{%
\setlength{\dimen0}{(#2-\widthof{#1})/2}%
\xdash{\dimen0}\text{#1}\xdash{\dimen0}}
% Math operators and abbreviations
\usepackage{forloop}
% Define \mcX = \mathcal{X}, with X = A-Z
% and \mbX = \mathbb{X}
\newcounter{ct}
\newcommand*\mc[1]{\mathcal{#1}}
\newcommand*\mb[1]{\mathbb{#1}}
\forloop{ct}{1}{\value{ct} < 27}{%
\expandafter\edef\csname mc\Alph{ct}\endcsname{\noexpand\mathcal{\Alph{ct}}}%
\expandafter\edef\csname mb\Alph{ct}\endcsname{\noexpand\mathbb{\Alph{ct}}}%
}
%
\newcommand*{\blank}{\raisebox{1pt}{\texttt{\char32}}}
\newcommand*\A{\mathcal{A}}
\newcommand*\Powerset{\mathcal{P}}
\newcommand*\dotcup{\mathrel{\dot\cup}}
\DeclareMathOperator\Konf{Konf}
\DeclareMathOperator\out{out}
\DeclareMathOperator\Sim{Sim}
\DeclareMathOperator\Instr{Instr}
\DeclareMathOperator\inc{inc}
\DeclareMathOperator\dec{dec}
\DeclareMathOperator\DTAPE{DTAPE}
\DeclareMathOperator\NTAPE{NTAPE}
\DeclareMathOperator\NTIME{NTIME}
\DeclareMathOperator\DTIME{DTIME}
\DeclareMathOperator\NSPACE{NSPACE}
\DeclareMathOperator\DSPACE{DSPACE}
\DeclareMathOperator\code{code}
\DeclareMathOperator\pos{pos}
\DeclareMathOperator\state{state}
\DeclareMathOperator\tape{tape}
\DeclareMathOperator\CLIQUE{CLIQUE}
\DeclareMathOperator\PREC{PREC} % Def. 8.2
\DeclareMathOperator\add{add} % Bsp. zu 8.2
\DeclareMathOperator\mult{mult} % Bsp. zu 8.2
% Faster compilation: one section / file.
%\includeonly{6-Berechenbarkeit}
%========================%
\begin{document}
\title{Vorlesung Informatik III -- Theoretische Informatik}
\subtitle{Formale Sprachen, Berechenbarkeit \& Komplexitätstheorie}
\author{Prof. Dr. Peter Thiemann\\\normalsize{Basierend auf einem Mitschrieb im WS\, 2015/16\ von Ralph Lesch\thanks{[email protected]}}}
\date{WS\,2016/17%\\2015-10-21 -- 2016-02-22
\\
\vspace{10ex}
\small\sffamily Zuletzt aktualisiert: \today}
\maketitle
\vspace{\baselineskip}
\tableofcontents
\include{1-Vorspann_Sprachen}
\include{2-Regulaere_Sprachen_und_endliche_Automaten}
\include{3-Grammatiken_und_kontextfreie_Sprachen}
\include{4-Kellerautomaten}
\include{5-Turing_und_Church}
\include{6-Berechenbarkeit}
\include{7-Komplexitaetstheorie}
% \include{8-Rekursive_Funktionen}
\renewcommand{\listtheoremname}{Liste der Definitionen}
\listoftheorems[ignoreall,show={Def,Def*,subDef},numwidth=2.8em]
\renewcommand{\listtheoremname}{Liste der Sätze}
\listoftheorems[ignoreall,show={Satz,Satz*,lemma,lemma*,Korollar,Korollar*},numwidth=2.8em]
\listoffigures
\addsec{Abkürzungsverzeichnis}
\begin{acronym}[MPCP]
\acro{AL}{Aussagenlogik}
\acro{CFL}{Menge der kontextfreien Sprachen}
\acro{CFG}{Menge der kontextfreien Grammatiken}
\acro{CNF}{Chomsky Normalform}
\acro{CP}{Korrespondenzproblem}
\acro{CYK}{Cocke, Younger, Kasami}
\acro{DAG}{gerichteter azyklischer Graph}
\acro{DCFG}{deterministische \acs*{CFG}}
\acro{DCFL}{deterministische \acs*{CFL}}
\acro{DEA}{deterministischer endlicher Automat}
\acro{DFA}{\acroextra{engl.: }deterministic finite automaton}
\acro{DPDA}{deterministischer Kellerautomat}
\acro{DTM}{deterministische \acs*{TM}}
\acro{EA}{endlicher Automat}
\acro{LBA}{Linear Bounded Automaton}
\acro{MPCP}{Das modifizierte \acs*{PCP}}
\acro{ND}{Nicht-Determinismus}
\acro{NEA}{nichtdeterministischer endlicher Automat}
\acro{NFA}{\acroextra{engl.: }nondeterministic finite automaton}
\acro{NPDA}{nichtdeterministischer Kellerautomat}
\acro{NT}{Nichtterminal}
\acroplural{NT}[NT]{Nichtterminale}
\acro{NTM}{Eine nichtdeterministische \acs*{TM}}
\acro{PCP}{Das Postsche Korrespondenzproblem}
\acro{PDA}{pushdown automaton\acroextra{ (Kellerautomat)}}
\acro{PL}{Pumping Lemma}
\acro{RE}[\textit{RE}]{Menge der regulären Ausdrücke}
\acro{REG}[\textit{REG}]{Menge der regulären Sprachen}
\acro{RM}{Registermaschine}
\acro{TM}{Turing-Maschine}
\acro{TT}{Turingtabelle}
\end{acronym}
\listoffixmes
\end{document}