e1.tex

   1 \documentclass{article}
   2
   3 \usepackage{a4wide}
   4 \usepackage{amssymb}
   5 \usepackage{amsmath}
   6
   7 \newcommand{\ra}{\rightarrow}
   8 \newcommand{\un}{\text{ \textsf{U} }}
   9 \newcommand{\ev}{\lozenge}
  10 \newcommand{\al}{\square}
  11 \newcommand{\nx}{\bigcirc}
  12
  13 \begin{document}
  14 \begin{enumerate}
  15         \item
  16         \item
  17         \item
  18         \item\strut\\
  19                 $\begin{array}{ll}
  20                         \phi_1=\ev\al c &
  21                                 s_2\ra s_4\overline{\ra s_5}\\
  22                         \phi_2=\al\ev c &
  23                                 s_2\ra s_4\overline{\ra s_5}\\
  24                         \phi_3=\nx\neg c\rightarrow\nx\nx c &
  25                                 s_2\ra s_4\overline{\ra s_5}\\
  26                         \phi_4=\al a &
  27                                 \text{Impossible. $a$ only holds in $s_1$ and in $s_5$.}\\
  28                         & \text{$s_1$ we can only leave through a state where $\neg a$}\\
  29                         & \text{$s_5$ we can not reach through $s_1$ without passing
  30                                 through a $\neg a$ state}\\
  31                         \phi_5=a\un\al(b\vee c) &
  32                                 s_1\ra s_4\overline{\ra s_5}\\
  33                         \phi_6=(\nx\nx b)\un(b\vee c) &
  34                                 s_2\ra s_4\overline{\ra s_5}
  35                 \end{array}$
  36 \end{enumerate}
  37 \end{document}