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 \everymath{\displaystyle}
  15
  16 \author{Mart Lubbers}
  17 \date{\today}
  18 \title{Model Checking Excercises 1}
  19
  20 \begin{document}
  21 \maketitle
  22 \begin{enumerate}
  23         \item
  24         \item
  25                 $\al(crit_0\veebar crit_1)$
  26         \item
  27                 We define:\\
  28                 $ AP=\bigcup_{i\in N}\{lift_n, door_n,call_n\},
  29                 N=\{0, 1, 2, 3\}$
  30                 \begin{enumerate}
  31                         \item
  32                                 $\bigwedge_{n\in N}\al(door_m\ra lift_n)$
  33                         \item
  34                                 $\bigwedge_{n\in N}\al(call_n\ra\ev(lift_n\wedge door_n))$
  35                         \item
  36                                 $\al\ev lift_0$
  37                         \item
  38                                 $\al(call_3\rightarrow\bigcirc (lift_3\wedge door_3))$
  39                 \end{enumerate}
  40         \item\strut\\
  41                 $\begin{array}{ll}
  42                         \varphi_1=\ev\al c &
  43                                 s_2\ra s_4\overline{\ra s_5}\\
  44                         \varphi_2=\al\ev c &
  45                                 s_2\ra s_4\overline{\ra s_5}\\
  46                         \varphi_3=\nx\neg c\rightarrow\nx\nx c &
  47                                 s_2\ra s_4\overline{\ra s_5}\\
  48                         \varphi_4=\al a &
  49                                 \text{Impossible. $a$ only holds in $s_1$ and in $s_5$.}\\
  50                         & \text{$s_1$ we can only leave through a state where $\neg a$}\\
  51                         & \text{$s_5$ we can not reach through $s_1$ without passing
  52                                 through a $\neg a$ state}\\
  53                         \varphi_5=a\un\al(b\vee c) &
  54                                 s_1\ra s_4\overline{\ra s_5}\\
  55                         \varphi_6=(\nx\nx b)\un(b\vee c) &
  56                                 s_2\ra s_4\overline{\ra s_5}
  57                 \end{array}$
  58 \end{enumerate}
  59 \end{document}