
sawine@55: \documentclass[9pt]{beamer}
sawine@55: \usetheme{Boadilla}
sawine@58: \usecolortheme{dove}
sawine@58: \usecolortheme{orchid}
sawine@58: \usecolortheme{dolphin}
sawine@58: %\usecolortheme{seagull}
sawine@58: 
sawine@60: \usepackage{amsmath, amsthm, amssymb, amsfonts, verbatim}
sawine@62: \usepackage{pifont}
sawine@61: \usepackage{xcolor}
sawine@58: \usepackage{ulem}
sawine@66: \usepackage{graphics}
sawine@58: \usepackage{tikz}
sawine@58: \usetikzlibrary{automata}
sawine@58: \usepackage{subfigure}
sawine@55: 
sawine@61: \renewcommand{\emph}{\textit}
sawine@66: \renewcommand{\em}{\it}
sawine@62: 
sawine@62: \newcommand{\cross}{\ding{55}}
sawine@55: \newcommand{\M}{\mathcal{M}}
sawine@55: \newcommand{\N}{\mathbb{N}_0}
sawine@55: \newcommand{\F}{\mathcal{F}}
sawine@57: \newcommand{\Fs}{\mathbb{F}}
sawine@55: \newcommand{\Prop}{\mathcal{P}}
sawine@55: \newcommand{\A}{\mathcal{A}}
sawine@55: \newcommand{\X}{\mathcal{X}}
sawine@55: \newcommand{\U}{\mathcal{U}}
sawine@55: \newcommand{\V}{\mathcal{V}}
sawine@55: \newcommand{\dnf}{\mathsf{dnf}}
sawine@55: \newcommand{\consq}{\mathsf{consq}}
sawine@55: 
sawine@55: \theoremstyle{definition} %plain, definition, remark, proof, corollary
sawine@55: \newtheorem*{def:finite words}{Finite words}
sawine@55: \newtheorem*{def:infinite words}{Infinite words}
sawine@55: \newtheorem*{def:regular languages}{Regular languages}
sawine@55: \newtheorem*{def:regular languages closure}{Regular closure}
sawine@55: \newtheorem*{def:omega regular languages}{$\omega$-regular languages}
sawine@55: \newtheorem*{def:omega regular languages closure}{$\omega$-regular closure}
sawine@55: \newtheorem*{def:buechi automata}{Automaton}
sawine@56: \newtheorem*{def:automata runs}{Run}
sawine@56: \newtheorem*{def:inf}{Infinite occurrences}
sawine@55: \newtheorem*{def:automata acceptance}{Acceptance}
sawine@56: \newtheorem*{def:automata language}{Recognised language}
sawine@57: \newtheorem*{def:general automata}{Generalised automaton}
sawine@55: \newtheorem*{def:general acceptance}{Acceptance}
sawine@61: \newtheorem*{def:syntax}{Syntax}
sawine@61: 
sawine@61: 
sawine@55: \newtheorem*{def:vocabulary}{Vocabulary}
sawine@62: \newtheorem*{def:frames}{Frame}
sawine@62: \newtheorem*{def:models}{Model}
sawine@55: \newtheorem*{def:satisfiability}{Satisfiability}
sawine@55: \newtheorem*{def:fs closure}{Closure}
sawine@55: \newtheorem*{def:atoms}{Atoms}
sawine@55: \newtheorem*{def:rep function}{Representation function}
sawine@55: \newtheorem*{def:next}{Next function}
sawine@55: \newtheorem*{def:dnf}{Disjunctive normal form}
sawine@55: \newtheorem*{def:positive formulae}{Positive formulae}
sawine@55: \newtheorem*{def:consq}{Logical consequences}
sawine@55: \newtheorem*{def:partial automata}{Partial automata}
sawine@55: 
sawine@55: \theoremstyle{plain}
sawine@55: \newtheorem{prop:vocabulary sat}{Proposition}[section]
sawine@55: \newtheorem{prop:general equiv}{Proposition}[section]
sawine@55: \newtheorem{prop:computation set=language}{Proposition}[section]
sawine@55: 
sawine@55: \theoremstyle{plain}
sawine@55: \newtheorem{thm:model language}{Theorem}[section]
sawine@55: \newtheorem{cor:mod=model language}{Corollary}[thm:model language]
sawine@55: \newtheorem{cor:mod=program language}[cor:mod=model language]{Corollary}
sawine@55: \newtheorem{thm:model checking}{Theorem}[section]
sawine@55: \newtheorem{lem:dnf}{Lemma}[section]
sawine@55: \newtheorem{lem:consq}[lem:dnf]{Lemma}
sawine@55: 
sawine@55: \title[Algorithmic Verification]{Algorithmic Verification of Reactive Systems}
sawine@55: \author{Eugen Sawin}
sawine@55: \institute[University of Freiburg]
sawine@55: { 
sawine@55:   Research Group for Foundations in Artificial Intelligence\\
sawine@55:   Computer Science Department\\
sawine@55:   University of Freiburg
sawine@55: }
sawine@55: \date[SS 2011]{Seminar: Automata Constructions in Model Checking}
sawine@55: \subject{Model Checking}
sawine@55: 
sawine@55: \begin{document}
sawine@55: \frame{\titlepage}
sawine@66: 
sawine@55: \begin{frame}
sawine@55: \frametitle{Motivation}
sawine@67: \framesubtitle{Model Checking 1/2}
sawine@66: \begin{center}
sawine@66: %\only<1>{\colorbox{black}{\makebox(35,10){\color{white} $\M \models \varphi$}}}
sawine@67: \[\M \models \varphi\]
sawine@66: \end{center}
sawine@66: \end{frame}
sawine@66: 
sawine@66: \begin{frame}
sawine@66: \frametitle{Motivation}
sawine@67: \framesubtitle{Model Checking 2/2}
sawine@67: \begin{center}
sawine@67: Given a program $P$ and specification $\varphi$:\\
sawine@67: \colorbox{black}{\makebox(150,10){\color{white} does every run of $P$ satisfy $\varphi$?}}
sawine@67: \end{center}
sawine@67: \end{frame}
sawine@67: 
sawine@67: \begin{frame}
sawine@67: \frametitle{Motivation}
sawine@67: \framesubtitle{Industry}
sawine@66: \begin{figure}
sawine@66: \centering
sawine@67: \subfigure{\includegraphics[width=70pt,height=50pt]{images/intel.jpg}}
sawine@67: \subfigure{\includegraphics[width=70pt,height=50pt]{images/airbag.jpg}}
sawine@67: \subfigure{\includegraphics[width=70pt,height=50pt]{images/atc.jpg}}
sawine@66: \end{figure}
sawine@55: \end{frame}
sawine@55: 
sawine@67: {
sawine@67: \setbeamercolor{normal text}{bg=black, fg=white}
sawine@67: \setbeamercolor{frametitle}{fg=white!30!black}
sawine@67: \usebeamercolor*{normal text} 
sawine@67: \usebeamercolor*{frametitle} 
sawine@67: \begin{frame}
sawine@67: \frametitle{Linear Temporal Logic}
sawine@70: \framesubtitle{Natural language 1/2}
sawine@67: \begin{center}
sawine@67: ``It is dark.''\\
sawine@67: \visible<2->{``It is \emph{always} dark.''\\}
sawine@67: \visible<3->{``It is \emph{currently} dark.''\\}
sawine@69: \visible<4->{``It will \emph{necessarily} be dark.''\\}
sawine@67: \visible<5->{``It is dark \emph{until} someone puts the light on.''}
sawine@67: \end{center}
sawine@67: \end{frame}
sawine@67: }
sawine@67: 
sawine@67: \begin{frame}
sawine@67: \frametitle{Linear Temporal Logic}
sawine@70: \framesubtitle{Natural language 2/2}
sawine@67: \begin{center}
sawine@67: \only<1->{
sawine@67: \color{white}
sawine@67: \colorbox{black}{\makebox(50,10){It is dark}} \colorbox{orange}{\makebox(30,10){until}} \colorbox{black!30}{\makebox(50,10){there is light}}\\
sawine@67: \visible<2->{
sawine@67: \colorbox{black}{\makebox(50,10){$p_0$}} \colorbox{orange}{\makebox(30,10){$\U$}} \colorbox{black!30}{\makebox(50,10){$p_1$}}}
sawine@67: }
sawine@67: \end{center}
sawine@67: \end{frame}
sawine@67: 
sawine@67: \begin{frame}
sawine@67: \frametitle{Linear Temporal Logic}
sawine@67: \framesubtitle{Syntax}
sawine@67: \begin{def:syntax}
sawine@67: Let $\Prop$ be the countable set of \emph{atomic propositions}, LTL-formulae $\varphi$ are defined using following productions:
sawine@67: \[\varphi ::= p \in \Prop \,|\, \neg \varphi \,|\, \varphi \lor \varphi \,|\, \X \varphi \,|\, \varphi \U \varphi\]
sawine@67: \begin{itemize}
sawine@67: \item $\neg, \lor$ corresponds to the Boolean \emph{negation} and \emph{disjunction}.
sawine@67: \item $\X$ reads \emph{next}.
sawine@67: \item $\U$ reads \emph{until}.
sawine@67: \end{itemize}
sawine@67: \end{def:syntax}
sawine@67: \end{frame}
sawine@67: 
sawine@67: \begin{frame}
sawine@67: \frametitle{Linear Temporal Logic}
sawine@67: \framesubtitle{Semantics}
sawine@67: \begin{def:frames}
sawine@67: An LTL-\emph{frame} is a tuple $\F = (S, R)$:
sawine@67: \begin{itemize}
sawine@67: \item $S = \{s_i \mid i \in \N\}$ is the set of states.
sawine@67: \item $R = \{(s_i, s_{i+1}) \mid i \in \N\}$ is the accessibility relation.
sawine@67: \end{itemize} 
sawine@67: \end{def:frames}
sawine@67: 
sawine@67: \begin{def:models}
sawine@67: An LTL-\emph{model} is a tuple $\M = (\F, V)$:
sawine@67: \begin{itemize}
sawine@67: \item $\F$ is a \emph{frame}.
sawine@67: \item $V: S \to 2^\Prop$ is a \emph{valuation function}.
sawine@67: \item Intuitively we say $p \in \Prop$ is \emph{true} at time instant $i$ iff $p \in V(i)$. 
sawine@67: \end{itemize}
sawine@67: \end{def:models}
sawine@67: \end{frame}
sawine@67: 
sawine@67: \begin{frame}
sawine@67: \frametitle{Linear Temporal Logic}
sawine@67: \framesubtitle{Model Example}
sawine@67: \begin{figure}
sawine@67: \centering
sawine@67: \begin{tikzpicture}[shorten >=1pt, node distance=2.5cm, auto, semithick, >=stealth   
sawine@67:     ,accepting/.style={fill, gray!50!black, text=white}]
sawine@67: \node[state, initial, initial text=] (s_0) {$\{p_0\}$};
sawine@67: \path (s_0) [late options={label=below:$s_0$}];
sawine@67: \node[state] (s_1) [right of= s_0] {$\{p_0, p_2\}$};
sawine@67: \path (s_1) [late options={label=below:$s_1$}];
sawine@67: \node[state] (s_2) [right of= s_1] {$\{p_1\}$};
sawine@67: \path (s_2) [late options={label=below:$s_2$}];
sawine@67: \node[state] (s_i) [right of= s_2] {$\{p_1\}$};
sawine@67: \path (s_i) [late options={label=below:$s_i$}];
sawine@67: \path[->] 
sawine@67: (s_0) edge node {$R$} (s_1) 
sawine@67: (s_1) edge node {$R$} (s_2);
sawine@67: \path[dashed,->] 
sawine@67: (s_2) edge node {$R$} (s_i); 
sawine@67: \end{tikzpicture}
sawine@67: \end{figure}
sawine@67: \end{frame}
sawine@67: 
sawine@67: \begin{frame}
sawine@67: \frametitle{Linear Temporal Logic}
sawine@67: \framesubtitle{Satisfiability}
sawine@67: \begin{def:satisfiability}
sawine@67: A model $\M = (\F, V)$ \emph{satisfies} a formula $\varphi$ at time instant $i$ is denoted by $\M,i \models \varphi$:
sawine@67: \begin{itemize}
sawine@67: \item $\M,i \models p$ for $p \in \Prop \iff p \in V(i)$
sawine@67: \item $\M,i \models \neg \varphi \iff$ not $\M,i \models \varphi$
sawine@67: \item $\M,i \models \varphi \lor \psi \iff \M,i \models \varphi$ or $\M,i \models \psi$
sawine@67: \item $\M,i \models \X \varphi \iff \M,i+1 \models \varphi$
sawine@67: \item $\M,i \models \varphi \U \psi \iff \exists{k \geq i}: \M,k \models \psi$ and $\forall{i \leq j < k}: \M,j \models\varphi$
sawine@67: \end{itemize}
sawine@67: \end{def:satisfiability}
sawine@67: \end{frame}
sawine@67: 
sawine@55: \begin{frame}
sawine@70: \frametitle{Reactive Systems}
sawine@70: \framesubtitle{Infinite inputs}
sawine@71: \begin{center}
sawine@66: \begin{figure}
sawine@71: \setcounter{subfigure}{0} 
sawine@71: \subfigure[Terminating program]{
sawine@70: \begin{tikzpicture}[shorten >=1pt, node distance=1.5cm, semithick, >=stealth   
sawine@70:     ,accepting/.style={fill, gray!50!black, text=white}]
sawine@71: \node[state, initial, initial text=$input$] (p) {$P$};
sawine@71: \coordinate (b) at (1,0);
sawine@70:         %\coordinate (b) at ($(a)+1/2*(3,3)$);
sawine@71: \draw (p) edge[->] node[right] {$\hspace{8pt}output$} (b);
sawine@70: %\draw[->] (p) -- (b);
sawine@70: \end{tikzpicture}
sawine@70: }
sawine@71: \hspace{10pt}
sawine@71: \visible<2>{\subfigure[Reactive program]{
sawine@66: \begin{tikzpicture}[shorten >=1pt, node distance=1.5cm, semithick, >=stealth   
sawine@66:     ,accepting/.style={fill, gray!50!black, text=white}]
sawine@71: \node[state, initial, initial text=$event$] (p) {$RP$};
sawine@71: \coordinate (b) at (1.1,0);
sawine@71:         %\coordinate (b) at ($(a)+1/2*(3,3)$);
sawine@71: \path[->]
sawine@71: (p) edge [loop right] node {$action$} ();
sawine@66: \end{tikzpicture}
sawine@71: }}
sawine@66: \end{figure}
sawine@71: \end{center}
sawine@55: \end{frame}
sawine@55: 
sawine@55: \begin{frame}
sawine@56: \frametitle{Automata}
sawine@59: \framesubtitle{Example 1/2}
sawine@58: \begin{figure}
sawine@58: \centering
sawine@60: \only<-3>{
sawine@58: \begin{tikzpicture}[shorten >=1pt, node distance=2cm, auto, semithick, >=stealth
sawine@58:     %every state/.style={fill, draw=none, gray, text=white},
sawine@58:     ,accepting/.style={fill, gray!50!black, text=white}
sawine@58:     %initial/.style ={gray, text=white}]%,  thick]
sawine@58:     ]
sawine@58: \node[state,initial, initial text=] (q_0) {$q_0$};
sawine@58: \node[state] (q_1) [above right of= q_0] {$q_1$};
sawine@58: \node[state,accepting](q_2) [below right of= q_1] {$q_2$};
sawine@58: \path[->] 
sawine@58: (q_0) edge node {$a$} (q_1)
sawine@58:   edge [loop above] node {$b$} ()
sawine@58: (q_1) edge node {$b$} (q_2)
sawine@58:   edge [loop above] node {$a$} ()
sawine@58: (q_2) %edge node {$a$} (q_1)
sawine@58:   edge node {$b$} (q_0);
sawine@58: \draw[->] (q_2) .. controls +(up:1cm) and +(right:1cm) .. node[above] {$a$} (q_1);
sawine@60: \end{tikzpicture}\\
sawine@60: \vspace{10pt}
sawine@60: \visible<2-3>{$w_1 = \overline{bbaa} \implies \rho_1 = q_0q_0\overline{q_0q_1q_1q_2}$}\\
sawine@60: \visible<3>{$w_2 = bb\overline{ab} \implies \rho_2 = q_0q_0\overline{q_1q_2}$}\\
sawine@60: \vspace{10pt}
sawine@60: \visible<4>{Accepts all inputs with infinite occurrences of $ab$.}
sawine@58: }
sawine@58: 
sawine@60: \only<4>{
sawine@58: \begin{tikzpicture}[shorten >=1pt, node distance=2cm, auto, semithick, >=stealth
sawine@58:     %every state/.style={fill, draw=none, gray, text=white},
sawine@58:     ,accepting/.style={fill, gray!50!black, text=white}
sawine@58:     %initial/.style ={gray, text=white}]%,  thick]
sawine@58:     ]
sawine@58: \node[state,initial, initial text=] (q_0) {$q_0$};
sawine@58: \node[state] (q_1) [above right of= q_0] {$q_1$};
sawine@58: \node[state,accepting](q_2) [below right of= q_1] {$q_2$};
sawine@58: \path[->] 
sawine@58: (q_0) 
sawine@58:   edge [loop above] node {$b$} ()
sawine@58: (q_1) 
sawine@58:   edge [loop above] node {$a$} ()
sawine@58: (q_2) %edge node {$a$} (q_1)
sawine@58:   edge node {$b$} (q_0);
sawine@58: \color{green}
sawine@58: \path[->] 
sawine@58: (q_0) edge node {$a$} (q_1) 
sawine@58: (q_1) edge node {$b$} (q_2);
sawine@58: \draw[->] (q_2) .. controls +(up:1cm) and +(right:1cm) .. node[above] {$a$} (q_1);
sawine@59: \end{tikzpicture}\\
sawine@58: \color{black}
sawine@60: \vspace{10pt}
sawine@60: \visible<2->{$w_1 = \overline{\textcolor{green}{b}ba\textcolor{green}{a}} \implies \rho_1 = q_0q_0\overline{q_0q_1q_1\textcolor{green}{q_2}}$}\\
sawine@60: \visible<3->{$w_2 = bb\overline{\textcolor{green}{ab}} \implies \rho_2 = q_0q_0\overline{q_1\textcolor{green}{q_2}}$}\\
sawine@60: \vspace{10pt}
sawine@60: \visible<4>{Accepts all inputs with infinite occurrences of $ab$.}
sawine@58: }
sawine@58: %Automaton $\A_1$
sawine@58: \end{figure}
sawine@58: \end{frame}
sawine@58: 
sawine@58: \begin{frame}
sawine@58: \frametitle{Automata}
sawine@59: \framesubtitle{Example 2/2 (Complement)}
sawine@58: \begin{figure}
sawine@58: \centering
sawine@60: \only<1>{
sawine@58:   \subfigure
sawine@58:             {
sawine@58:               \begin{tikzpicture}[shorten >=1pt, node distance=2cm, auto, semithick, >=stealth
sawine@58:                   %every state/.style={fill, draw=none, gray, text=white},
sawine@58:                   ,accepting/.style={fill, gray!50!black, text=white}
sawine@58:                   %initial/.style ={gray, text=white}]%,  thick]
sawine@58:                 ]
sawine@58:                 \node[state,initial, initial text=, accepting] (q_0) {$q_0$};
sawine@58:                 \node[state, accepting] (q_1) [above right of= q_0] {$q_1$};
sawine@58:                 \node[state](q_2) [below right of= q_1] {$q_2$};
sawine@58:                 \path[->] 
sawine@58:                 (q_0) edge node {$a$} (q_1)
sawine@58:                 edge [loop above] node {$b$} ()
sawine@58:                 (q_1) edge node {$b$} (q_2)
sawine@58:                 edge [loop above] node {$a$} ()
sawine@58:                 (q_2) %edge node {$a$} (q_1)
sawine@58:                 edge node {$b$} (q_0);
sawine@58:                 \draw[->] (q_2) .. controls +(up:1cm) and +(right:1cm) .. node[above] {$a$} (q_1);
sawine@58:               \end{tikzpicture}
sawine@58:             }
sawine@58: }
sawine@58: \only<2>{ 
sawine@58:   \subfigure
sawine@58:             {
sawine@58:               \begin{tikzpicture}[shorten >=1pt, node distance=2cm, auto, semithick, >=stealth
sawine@58:                   %every state/.style={fill, draw=none, gray, text=white},
sawine@58:                   ,accepting/.style={fill, gray!50!black, text=white}
sawine@58:                   %initial/.style ={gray, text=white}]%,  thick]
sawine@58:                 ]
sawine@58:                 \node[state,initial, initial text=, accepting] (q_0) {$q_0$};
sawine@58:                 \node[state, accepting] (q_1) [above right of= q_0] {$q_1$};
sawine@58:                 \node[state](q_2) [below right of= q_1] {$q_2$};
sawine@58:                 \path[->] 
sawine@58:                 (q_0) 
sawine@58:                 edge [loop above] node {$b$} ()
sawine@58:                 (q_1) 
sawine@58:                 edge [loop above] node {$a$} ();                
sawine@58:                 \color{red}  
sawine@58:                 \path[->] 
sawine@58:                 (q_0) edge node {$a$} (q_1)
sawine@58:                 (q_1) edge node {$b$} (q_2)
sawine@58:                 (q_2) edge node {$b$} (q_0);  
sawine@58:                 \draw[->] (q_2) .. controls +(up:1cm) and +(right:1cm) .. node[above] {$a$} (q_1);              
sawine@58:               \end{tikzpicture}  \color{red}  
sawine@58:             }
sawine@58:  \color{black}
sawine@58: }
sawine@58: \only<3->{ \setcounter{subfigure}{0} 
sawine@62:   \subfigure[Complement automaton \cross]
sawine@58:             {
sawine@58:               \begin{tikzpicture}[shorten >=1pt, node distance=2cm, auto, semithick, >=stealth
sawine@58:                   %every state/.style={fill, draw=none, gray, text=white},
sawine@58:                   ,accepting/.style={fill, gray!50!black, text=white}
sawine@58:                   %initial/.style ={gray, text=white}]%,  thick]
sawine@58:                 ]
sawine@58:                 \node[state,initial, initial text=, accepting] (q_0) {$q_0$};
sawine@58:                 \node[state, accepting] (q_1) [above right of= q_0] {$q_1$};
sawine@58:                 \node[state](q_2) [below right of= q_1] {$q_2$};
sawine@58:                 \path[->] 
sawine@58:                 (q_0) 
sawine@58:                 edge [loop above] node {$b$} ()
sawine@58:                 (q_1) 
sawine@58:                 edge [loop above] node {$a$} ();                
sawine@58:                 \path[->] 
sawine@58:                 (q_0) edge node {$a$} (q_1)
sawine@58:                 (q_1) edge node {$b$} (q_2)
sawine@58:                 (q_2) edge node {$b$} (q_0);  
sawine@58:                 \draw[->] (q_2) .. controls +(up:1cm) and +(right:1cm) .. node[above] {$a$} (q_1);              
sawine@58:               \end{tikzpicture}  \color{red}  
sawine@58:             }
sawine@58:  \color{black}
sawine@58: }
sawine@60: %\hspace{10pt}
sawine@58: \visible<3->{
sawine@62:   \subfigure[Complement automaton \checkmark]
sawine@58:             {
sawine@58:               \label{fig:complement automaton}
sawine@58:               \begin{tikzpicture}[shorten >=1pt, node distance=2cm, auto, semithick, >=stealth   
sawine@58:                   ,accepting/.style={fill, gray!50!black, text=white}]
sawine@58:                 \node[state, initial, initial text=] (q_0) {$q_0$};
sawine@58:                 \node[state, accepting] (q_1) [above right of= q_0] {$q_1$};
sawine@58:                 \node[state, accepting](q_2) [below right of= q_1] {$q_2$};
sawine@58:                 \path[->] 
sawine@58:                 (q_0) edge node {$a$} (q_1)
sawine@58:                 edge node {$b$} (q_2)
sawine@58:                 edge [loop above] node {$a, b$} ()
sawine@58:                 (q_1) edge [loop above] node {$a$} ()
sawine@58:                 (q_2) 
sawine@58:                 edge [loop above] node {$b$} ();
sawine@58:               \end{tikzpicture}\color{green}  
sawine@59:             }\\
sawine@58: \color{black}  
sawine@59: \vspace{20pt}
sawine@66: Accepts all inputs with finite many $ab$.
sawine@58: }
sawine@58: %\caption{Automata from Example \ref{ex:automaton}}
sawine@58: \end{figure}
sawine@58: \end{frame}
sawine@61: \color{black}  
sawine@58: 
sawine@58: \begin{frame}
sawine@58: \frametitle{Automata}
sawine@56: \framesubtitle{Definition}
sawine@55: \begin{def:buechi automata}
sawine@56: A non-deterministic B\"uchi automaton is a tuple $\A = (\Sigma, S, S_0, \Delta, F)$ with:
sawine@55: \begin{itemize}
sawine@57: \item $\Sigma$ is a finite \emph{alphabet}.
sawine@57: \item $S$ is a finite set of \emph{states}.
sawine@57: \item $S_0 \subseteq S$ is the set of \emph{initial states}.
sawine@57: \item $\Delta: S \times \Sigma \times S$ is a \emph{transition relation}.
sawine@57: \item $F \subseteq S$ is the set of \emph{accepting states}.
sawine@55: \end{itemize}
sawine@55: \end{def:buechi automata}
sawine@55: \end{frame}
sawine@55: 
sawine@55: \begin{frame}
sawine@56: \frametitle{Automata}
sawine@56: \framesubtitle{Runs}
sawine@56: \begin{def:automata runs}
sawine@56: Let $\A = (\Sigma, S, S_0, \Delta, F)$ be an automaton:
sawine@56: \begin{itemize}
sawine@56: \item A run $\rho$ of $\A$ on an infinite word $w = a_0,a_1,...$ is a sequence $\rho = s_0,s_1,...$:
sawine@56: \begin{itemize}
sawine@57: \item $s_0 \in S_0$.
sawine@57: \item $(s_i, a_i, s_{i+1}) \in \Delta$, for all $i \geq 0$.
sawine@56: \end{itemize}
sawine@57: \item Alternative view of a run $\rho$ as a function $\rho : \N \to S$, with $\rho(i) = s_i$.
sawine@56: \end{itemize}
sawine@56: \end{def:automata runs}
sawine@66: \visible<2->{\[w_1 = \overline{\textcolor{green}{b}ba\textcolor{green}{a}} \implies \rho_1 = q_0q_0\overline{q_0q_1q_1\textcolor{green}{q_2}}\]
sawine@66: \[w_2 = bb\overline{\textcolor{green}{ab}} \implies \rho_2 = q_0q_0\overline{q_1\textcolor{green}{q_2}}\]}
sawine@56: \end{frame}
sawine@56: 
sawine@56: \begin{frame}
sawine@56: \frametitle{Automata}
sawine@56: \framesubtitle{Acceptance}
sawine@56: \begin{def:inf}
sawine@56: Let $\A = (\Sigma, S, S_0, \Delta, F)$ be an automaton and let $\rho$ be a run of $\A$:
sawine@56: \begin{itemize}
sawine@59: \item $\exists^\omega$ denotes the existential quantifier for \emph{infinitely} many occurrences.
sawine@57: \item $inf(\rho) = \{s \in S \mid \exists^\omega{n \in \N}: \rho(n) = s\}$.
sawine@56: \end{itemize}
sawine@56: \end{def:inf}
sawine@56: 
sawine@56: \begin{def:automata acceptance}
sawine@56: Let $\A = (\Sigma, S, S_0, \Delta, F)$ be an automaton and let $\rho$ be a run of $\A$:
sawine@56: \begin{itemize}
sawine@56: \item $\rho$ is \emph{accepting} iff $inf(\rho) \cap F \neq \emptyset$.
sawine@59: \item $\A$ \emph{accepts} an input word $w$ iff there exists a run $\rho$ of $\A$ on $w$, such that $\rho$ is accepting. 
sawine@56: \end{itemize}
sawine@56: \end{def:automata acceptance}
sawine@56: \end{frame}
sawine@56: 
sawine@56: \begin{frame}
sawine@56: \frametitle{Automata}
sawine@56: \framesubtitle{Language}
sawine@56: \begin{def:automata language}
sawine@56: Let $\A = (\Sigma, S, S_0, \Delta, F)$ be an automaton:
sawine@56: \begin{itemize}
sawine@57: \item $L_\omega(\A) = \{w \in \Sigma^\omega \mid \A \text{ accepts } w\}$, we say $\A$ recognises language $L_\omega(\A)$.
sawine@57: \item Language $L$ is \emph{B\"uchi-recognisable} iff there is an automaton $\A$ with $L = L_\omega(\A)$.
sawine@56: \end{itemize}
sawine@56: \end{def:automata language}
sawine@56: \end{frame}
sawine@56: 
sawine@56: \begin{frame}
sawine@57: \frametitle{Generalised Automata}
sawine@57: \begin{def:general automata}
sawine@65: A \emph{generalised B\"uchi automaton} is a tuple $\A_G = (\Sigma, S, S_0, \Delta, \{F_i\}_{i<k})$:
sawine@57: \begin{itemize}
sawine@57: \item $\{F_i\}$ is a finite set of sets for a given $k$.
sawine@57: \item Each $F_i \subseteq S$ is a finite set of accepting states.
sawine@57: \end{itemize}
sawine@57: \end{def:general automata}
sawine@57: 
sawine@57: \begin{def:general acceptance}
sawine@65: Let $\A_G = (\Sigma, S, S_0, \Delta, \{F_i\}_{i<k})$ be a generalised automaton and let $\rho$ be a run of $\A_G$:
sawine@57: \begin{itemize}
sawine@57: \item $\rho$ is \emph{accepting} iff $\forall{i < k}: inf(\rho) \cap F_i \neq \emptyset$.
sawine@65: \item $\A_G$ \emph{accepts} an input word $w$ iff there exists a run $\rho$ of $\A_G$ on $w$, such that $\rho$ is accepting. 
sawine@57: \end{itemize} 
sawine@57: \end{def:general acceptance}
sawine@57: 
sawine@57: \begin{prop:general equiv}
sawine@65: Let $\A_G = (\Sigma, S, S_0, \Delta, \{F_i\}_{i < k})$ be a generalised automaton and let $\A_i = (\Sigma, S, S_0, \Delta, F_i)$ be non-deterministic automata:
sawine@65: \[L_\omega(\A_G) = \bigcap_{i < k} L_\omega(\A_i)\]
sawine@57: \end{prop:general equiv}
sawine@57: \end{frame}
sawine@57: 
sawine@65: \begin{frame}
sawine@68: \frametitle{Automata Construction}
sawine@68: \framesubtitle{Formula automata}
sawine@68: \begin{center}
sawine@68: Model $\M_\varphi$ for formula $\varphi$\\
sawine@68: $\Downarrow$\\
sawine@68: Closure $CL(\varphi)$ of $\varphi$\\
sawine@68: $\Downarrow$\\
sawine@68: Automaton $\A_\varphi$ for $\varphi$\\
sawine@68: \vspace{20pt}
sawine@68: \textcolor{red}{On-the-fly methods} \`a la Gerth et al.
sawine@68: \end{center}
sawine@68: \end{frame}
sawine@68: 
sawine@68: \begin{frame}
sawine@68: \frametitle{Automata Construction}
sawine@68: \framesubtitle{System automata}
sawine@68: \begin{center}
sawine@68: Model $\M_\varphi$ for formula $\varphi$\\
sawine@68: $\Downarrow$\\
sawine@68: Closure $CL(\varphi)$ of $\varphi$\\
sawine@68: $\Downarrow$\\
sawine@68: Automaton $\A_\varphi$ for $\varphi$
sawine@68: \end{center}
sawine@68: \end{frame}
sawine@68: 
sawine@68: \begin{frame}
sawine@55: \frametitle{Model Checking}
sawine@67: \framesubtitle{Definition}
sawine@66: \begin{thm:model checking}
sawine@66: \label{thm:model checking}
sawine@66: Let $\A_P$ be the automaton for program $P$ and let $\A_\varphi$ be the automaton for formula $\varphi$.\\
sawine@66: P satisfies $\varphi$ iff:
sawine@66: \begin{itemize}
sawine@66: \item $L_\omega(\A_P) \subseteq L_\omega(\A_\varphi)$.
sawine@66: \item $L_\omega(\A_P) \cap L_\omega(\A_{\neg \varphi}) = \emptyset$.
sawine@66: \end{itemize}
sawine@66: \end{thm:model checking}
sawine@55: \end{frame}
sawine@55: 
sawine@55: \begin{frame}[allowframebreaks]
sawine@55: \frametitle<presentation>{Literature}    
sawine@55: \begin{thebibliography}{10}    
sawine@55: 
sawine@55: \beamertemplatearticlebibitems
sawine@55: \bibitem{ref:ltl&büchi}
sawine@55: Madhavan Mukund.
sawine@55: \newblock {\em Linear-Time Temporal Logic and B\"uchi Automata}.
sawine@55: \newblock Winter School on Logic and Computer Science, Indian Statistical Institute, Calcutta, 1997.
sawine@55:   
sawine@55: \beamertemplatearticlebibitems
sawine@55: \bibitem{ref:alternating verification}
sawine@55: Moshe Y. Vardi.
sawine@55: \newblock {\em Alternating Automata and Program Verification}.
sawine@55: \newblock Computer Science Today, Volume 1000 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1995.
sawine@55: 
sawine@55: \beamertemplatebookbibitems
sawine@55: \bibitem{ref:handbook}
sawine@55: Patrick Blackburn and Frank Wolter and Johan van Benthem.
sawine@55: \newblock {\em Handbook of Modal Logic}.
sawine@55: \newblock 3rd Edition, Elsevier, Amsterdam, Chapter 11 P. 655-720 and Chapter 17 P. 975-989, 2007.
sawine@55: 
sawine@55: \end{thebibliography}
sawine@55: \end{frame}
sawine@55: 
sawine@55: \end{document}




