e2.tex

   1 \documentclass{article}
   2
   3 \usepackage{a4wide}
   4 \usepackage{amssymb}
   5 \usepackage{amsmath}
   6 \usepackage{booktabs}
   7 \usepackage[all]{xypic}
   8
   9 \newcommand{\ra}{\rightarrow}
  10 \newcommand{\un}{\text{ \textsf{U} }}
  11 \newcommand{\ev}{\lozenge}
  12 \newcommand{\al}{\square}
  13 \newcommand{\nx}{\bigcirc}
  14
  15 \everymath{\displaystyle}
  16
  17 \author{Mart Lubbers}
  18 \date{\today}
  19 \title{Model Checking Excercises 2}
  20
  21 \begin{document}
  22 \maketitle
  23 \begin{enumerate}
  24         \item\strut\\
  25                 \begin{tabular}{ll}
  26                         $TS\nvDash\varphi_1$ & $s_1\ra \overline{s_3\ra s_4}$\\
  27                         $TS\vDash\varphi_2$ & Yes, all loops go via $s_2,s_3$ or $s_5$
  28                         where $c$ holds.\\
  29                         $TS\vDash\varphi_3$ & Yes, $\nx\neg c$ is only true when the next
  30                         state is $s_4$.\\
  31                         & From $s_4$ you can only go to a state where $c$ holds.\\
  32                         $TS\nvDash\varphi_4$ & $s_2\ra\ldots$\\
  33                         $TS\vDash\varphi_5$ & Yes, starting in $s_1$ $b\vee c$ holds after
  34                         the first transition\\
  35                         & Starting in $s_2$ $b\vee c$ holds immediatly.\\
  36                         $TS\vDash\varphi_6$ & $s_1\ra\overline{s_4\ra s_2}$
  37                 \end{tabular}
  38         \item
  39                 \begin{equation}
  40                         \xymatrix{
  41                                 \ar[r] &
  42                                 *+[F]{s_0}\ar@/^/[r]^{\neg a}\ar@(d,l)^{a} &
  43                                 *+[F]{s_1}\ar@/^/[l]^{\neg b}\ar[r]^{b} &
  44                                 *+[Fo]{s_2}\ar@(d,r)_{true}
  45                         }
  46                 \end{equation}
  47
  48                 \begin{equation}
  49                         \xymatrix{
  50                                 \ar[r] &
  51                                 *+[Fo]{s_0}\ar@(d,l)^{true}\ar[r]^{a} &
  52                                 *+[F]{s_1}\ar@(d,l)^{true}\\
  53                                 \ar[r] & *+[F]{s_2}\ar@(d,l)^{\emptyset}\ar[r]^{a\vee b} &
  54                                 *+[Fo]{s_3}\ar@(d,l)^{true}
  55                         }
  56                 \end{equation}
  57
  58                 \begin{equation}
  59                         \xymatrix{
  60                                 \ar[r] &
  61                                 *+[Fo]{s_0}\ar[r]^{\neg a}\ar[d]_{a} &
  62                                 *+[F]{s_1}\ar@(d,r)_{true}\\
  63                                 & *+[Fo]{s_2}\ar@(d,l)^{true}
  64                         }
  65                 \end{equation}
  66         \item
  67                 $\ev\al(a\wedge\ev b)$
  68 \end{enumerate}
  69 \end{document}