e1.tex

   1 \documentclass{article}
   2
   3 \usepackage{a4wide}
   4 \usepackage{amssymb}
   5 \usepackage{amsmath}
   6 \usepackage[all]{xypic}
   7
   8 \newcommand{\ra}{\rightarrow}
   9 \newcommand{\un}{\text{ \textsf{U} }}
  10 \newcommand{\ev}{\lozenge}
  11 \newcommand{\al}{\square}
  12 \newcommand{\nx}{\bigcirc}
  13
  14 \begin{document}
  15 \begin{enumerate}
  16         \item
  17                 $\xymatrix{
  18                         \ar[dr]\\
  19                                 & *+[o][F-]{s_0}\\
  20                         *+[o][F-]{nc_1} & & *+[o][F-]{nc_1}
  21                 }$
  22         \item
  23                 $\al(crit_0\veebar crit_1)$
  24         \item
  25                 We define:\\
  26                 $\displaystyle AP=\bigcup_{i\in N}\{lift_n, door_n,call_n\},
  27                 N=\{0, 1, 2, 3\}$
  28                 \begin{enumerate}
  29                         \item
  30                                 $\forall n\in N\al(door_m\ra lift_n)$
  31                         \item
  32                                 $\forall n\in N\al(call_n\ra\ev lift_n)$
  33                         \item
  34                                 $\al\ev lift_0$
  35                         \item
  36                                 $\al call_3\rightarrow\bigcirc lift_3$
  37                 \end{enumerate}
  38         \item\strut\\
  39                 $\begin{array}{ll}
  40                         \varphi_1=\ev\al c &
  41                                 s_2\ra s_4\overline{\ra s_5}\\
  42                         \varphi_2=\al\ev c &
  43                                 s_2\ra s_4\overline{\ra s_5}\\
  44                         \varphi_3=\nx\neg c\rightarrow\nx\nx c &
  45                                 s_2\ra s_4\overline{\ra s_5}\\
  46                         \varphi_4=\al a &
  47                                 \text{Impossible. $a$ only holds in $s_1$ and in $s_5$.}\\
  48                         & \text{$s_1$ we can only leave through a state where $\neg a$}\\
  49                         & \text{$s_5$ we can not reach through $s_1$ without passing
  50                                 through a $\neg a$ state}\\
  51                         \varphi_5=a\un\al(b\vee c) &
  52                                 s_1\ra s_4\overline{\ra s_5}\\
  53                         \varphi_6=(\nx\nx b)\un(b\vee c) &
  54                                 s_2\ra s_4\overline{\ra s_5}
  55                 \end{array}$
  56 \end{enumerate}
  57 \end{document}