1.1 --- a/slides/src/slides.tex Thu Jul 21 01:41:30 2011 +0200
1.2 +++ b/slides/src/slides.tex Thu Jul 21 14:52:03 2011 +0200
1.3 @@ -325,41 +325,31 @@
1.4 \begin{itemize}
1.5 \item $L_\omega(\A) = \{w \in \Sigma^\omega \mid \A \text{ accepts } w\}$, we say $\A$ recognises language $L_\omega(\A)$.
1.6 \item Language $L$ is \emph{B\"uchi-recognisable} iff there is an automaton $\A$ with $L = L_\omega(\A)$.
1.7 -\item The class of B\"uchi-recognisable languages corresponds to the class of $\omega$-regular languages.
1.8 \end{itemize}
1.9 \end{def:automata language}
1.10 \end{frame}
1.11
1.12 \begin{frame}
1.13 \frametitle{Generalised Automata}
1.14 -\framesubtitle{Definition}
1.15 \begin{def:general automata}
1.16 -A \emph{generalised B\"uchi automaton} is a tuple $\A = (\Sigma, S, S_0, \Delta, \{F_i\}_{i<k})$:
1.17 +A \emph{generalised B\"uchi automaton} is a tuple $\A_G = (\Sigma, S, S_0, \Delta, \{F_i\}_{i<k})$:
1.18 \begin{itemize}
1.19 \item $\{F_i\}$ is a finite set of sets for a given $k$.
1.20 \item Each $F_i \subseteq S$ is a finite set of accepting states.
1.21 \end{itemize}
1.22 \end{def:general automata}
1.23 -\end{frame}
1.24
1.25 -\begin{frame}
1.26 -\frametitle{Generalised Automata}
1.27 -\framesubtitle{Acceptance}
1.28 \begin{def:general acceptance}
1.29 -Let $\A = (\Sigma, S, S_0, \Delta, \{F_i\}_{i<k})$ be a generalised automaton and let $\rho$ be a run of $\A$:
1.30 +Let $\A_G = (\Sigma, S, S_0, \Delta, \{F_i\}_{i<k})$ be a generalised automaton and let $\rho$ be a run of $\A_G$:
1.31 \begin{itemize}
1.32 \item $\rho$ is \emph{accepting} iff $\forall{i < k}: inf(\rho) \cap F_i \neq \emptyset$.
1.33 -\item $\A$ \emph{accepts} an input word $w$ iff there exists a run $\rho$ of $\A$ on $w$, such that $\rho$ is accepting.
1.34 +\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.
1.35 \end{itemize}
1.36 \end{def:general acceptance}
1.37 -\end{frame}
1.38
1.39 -\begin{frame}
1.40 -\frametitle{Generalised Automata}
1.41 -\framesubtitle{Language}
1.42 \begin{prop:general equiv}
1.43 -Let $\A = (\Sigma, S, S_0, \Delta, \{F_i\}_{i < k})$ be a generalised automaton and let $\A_i = (\Sigma, S, S_0, \Delta, F_i)$ be non-deterministic automata, then following equivalence condition holds:
1.44 -\[L_\omega(\A) = \bigcap_{i < k} L_\omega(\A_i)\]
1.45 +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:
1.46 +\[L_\omega(\A_G) = \bigcap_{i < k} L_\omega(\A_i)\]
1.47 \end{prop:general equiv}
1.48 \end{frame}
1.49
1.50 @@ -431,21 +421,6 @@
1.51
1.52 \begin{frame}
1.53 \frametitle{Linear Temporal Logic}
1.54 -\framesubtitle{Satisfiability}
1.55 -\begin{def:satisfiability}
1.56 -A model $\M = (\F, V)$ \emph{satisfies} a formula $\varphi$ at time instant $i$ is denoted by $\M,i \models \varphi$:
1.57 -\begin{itemize}
1.58 -\item $\M,i \models p$ for $p \in \Prop \iff p \in V(i)$
1.59 -\item $\M,i \models \neg \varphi \iff$ not $\M,i \models \varphi$
1.60 -\item $\M,i \models \varphi \lor \psi \iff \M,i \models \varphi$ or $\M,i \models \psi$
1.61 -\item $\M,i \models \X \varphi \iff \M,i+1 \models \varphi$
1.62 -\item $\M,i \models \varphi \U \psi \iff \exists{k \geq i}: \M,k \models \psi$ and $\forall{i \leq j < k}: \M,j \models\varphi$
1.63 -\end{itemize}
1.64 -\end{def:satisfiability}
1.65 -\end{frame}
1.66 -
1.67 -\begin{frame}
1.68 -\frametitle{Linear Temporal Logic}
1.69 \framesubtitle{Model Example}
1.70 \begin{figure}
1.71 \centering
1.72 @@ -469,6 +444,21 @@
1.73 \end{frame}
1.74
1.75 \begin{frame}
1.76 +\frametitle{Linear Temporal Logic}
1.77 +\framesubtitle{Satisfiability}
1.78 +\begin{def:satisfiability}
1.79 +A model $\M = (\F, V)$ \emph{satisfies} a formula $\varphi$ at time instant $i$ is denoted by $\M,i \models \varphi$:
1.80 +\begin{itemize}
1.81 +\item $\M,i \models p$ for $p \in \Prop \iff p \in V(i)$
1.82 +\item $\M,i \models \neg \varphi \iff$ not $\M,i \models \varphi$
1.83 +\item $\M,i \models \varphi \lor \psi \iff \M,i \models \varphi$ or $\M,i \models \psi$
1.84 +\item $\M,i \models \X \varphi \iff \M,i+1 \models \varphi$
1.85 +\item $\M,i \models \varphi \U \psi \iff \exists{k \geq i}: \M,k \models \psi$ and $\forall{i \leq j < k}: \M,j \models\varphi$
1.86 +\end{itemize}
1.87 +\end{def:satisfiability}
1.88 +\end{frame}
1.89 +
1.90 +\begin{frame}
1.91 \frametitle{Model Checking}
1.92 \framesubtitle{A bit more information about this}
1.93