
%\documentstyle[11pt,zed]{csart}
\documentstyle{csart}

\begin{document}

\sloppy

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\def\parallel{\mathinner{\,\|\,}}

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\newcommand{\proc}[1]{\mbox{\bf {#1}}}

\def\lbag{[[}
\def\rbag{]]}

%\section{Problem}

% With old labelling of channel states...
%\begin{displaymath}
%\begin{array}{lclclcl}
%\proc{C}^{00} (i, j) & = & \proc{P}^2 (i - 1) & \parallel &
%			   \proc{Q}^2 (j - 1) & \parallel &
%   \proc{T} \left( \lbag Q_{j - 1}.c.P_i, P_{i - 1}.c.Q_j \rbag \right) \\
%\proc{C}^{01} (i, j) & = & \proc{P}^5 (i - 1, j - 1) & \parallel &
%			   \proc{Q}^2 (j - 1) & \parallel &
%   \proc{T} \left( \lbag Q_{j - 1}.c.P_i \rbag \right) \\
%\proc{C}^{02} (i, j) & = & \proc{P}^0 (i) & \parallel &
%			   \proc{Q}^2 (j - 1) & \parallel &
%   \proc{T} \left( \lbag Q_{j - 1}.c.P_i, P_i.r.Q_j \rbag \right) \\
%\proc{C}^{03} (i, j) & = & \proc{P}^1 (i, j) & \parallel &
%			   \proc{Q}^2 (j - 1) & \parallel &
%   \proc{T} \left( \lbag Q_{j - 1}.c.P_i \rbag \right) \\
%\proc{C}^{04} (i, j) & = & \proc{P}^2 (i) & \parallel &
%			   \proc{Q}^2 (j - 1) & \parallel &
%   \proc{T} \left( \lbag Q_{j - 1}.c.P_i, Q_j.c.P_{i + 1} \rbag \right) \\
%\proc{C}^{05} (i, j) & = & \proc{P}^2 (i) & \parallel &
%			   \proc{Q}^6 (j - 1, i) & \parallel &
%   \proc{T} \left( \lbag Q_{j - 1}.c.P_i \rbag \right) \\
%\\
%\proc{C}^{10} (i, j) & = & \proc{P}^2 (i - 1) & \parallel &
%			   \proc{Q}^5 (j - 1, i - 1) & \parallel &
%   \proc{T} \left( \lbag P_{i - 1}.c.Q_j \rbag \right) \\
%\proc{C}^{11} (i, j) & = & \proc{P}^5 (i - 1, j - 1) & \parallel &
%			   \proc{Q}^5 (j - 1, i - 1) & \parallel &
%   \proc{T} \left( \lbag \rbag \right) \\
%\proc{C}^{12} (i, j) & = & \proc{P}^0 (i) & \parallel &
%			   \proc{Q}^5 (j - 1, i - 1) & \parallel &
%   \proc{T} \left( \lbag P_i.r.Q_j \rbag \right) \\
%\proc{C}^{13} (i, j) & = & \proc{P}^1 (i, j) & \parallel &
%			   \proc{Q}^5 (j - 1, i - 1) & \parallel &
%   \proc{T} \left( \lbag \rbag \right) \\
%\proc{C}^{14} (i, j) & = & \proc{P}^2 (i) & \parallel &
%			   \proc{Q}^5 (j - 1, i - 1) & \parallel &
%   \proc{T} \left( \lbag Q_j.c.P_{i + 1} \rbag \right) \\
%\proc{C}^{15} (i, j) & = & \proc{P}^2 (i) & \parallel &
%			   \proc{Q}^7 (j - 1, i) & \parallel &
%   \proc{T} \left( \lbag \rbag \right) \\
%\\
%\proc{C}^{20} (i, j) & = & \proc{P}^2 (i - 1) & \parallel &
%			   \proc{Q}^0 (j) & \parallel &
%   \proc{T} \left( \lbag Q_j.r.P_i, P_{i - 1}.c.Q_j \rbag \right) \\
%\proc{C}^{21} (i, j) & = & \proc{P}^5 (i - 1, j - 1) & \parallel &
%			   \proc{Q}^0 (j) & \parallel &
%   \proc{T} \left( \lbag Q_j.r.P_i \rbag \right) \\
%\proc{C}^{22} (i, j) & = & \proc{P}^0 (i) & \parallel &
%			   \proc{Q}^0 (j) & \parallel &
%   \proc{T} \left( \lbag Q_j.r.P_i, P_i.r.Q_j \rbag \right) \\
%\proc{C}^{23} (i, j) & = & \proc{P}^1 (i, j) & \parallel &
%			   \proc{Q}^0 (j) & \parallel &
%   \proc{T} \left( \lbag Q_j.r.P_i \rbag \right) \\
%\proc{C}^{24} (i, j) & = & \proc{P}^2 (i) & \parallel &
%			   \proc{Q}^0 (j) & \parallel &
%   \proc{T} \left( \lbag Q_j.r.P_i, Q_j.c.P_{i + 1} \rbag \right) \\
%\proc{C}^{25} (i, j) & = & \proc{P}^2 (i) & \parallel &
%			   \proc{Q}^3 (j, i) & \parallel &
%   \proc{T} \left( \lbag Q_j.r.P_i \rbag \right) \\
%\\
%\proc{C}^{30} (i, j) & = & \proc{P}^2 (i - 1) & \parallel &
%			   \proc{Q}^1 (j, i) & \parallel &
%   \proc{T} \left( \lbag P_{i - 1}.c.Q_j \rbag \right) \\
%\proc{C}^{31} (i, j) & = & \proc{P}^5 (i - 1, j - 1) & \parallel &
%			   \proc{Q}^1 (j, i) & \parallel &
%   \proc{T} \left( \lbag \rbag \right) \\
%\proc{C}^{32} (i, j) & = & \proc{P}^0 (i) & \parallel &
%			   \proc{Q}^1 (j, i) & \parallel &
%   \proc{T} \left( \lbag P_i.r.Q_j \rbag \right) \\
%\proc{C}^{33} (i, j) & = & \proc{P}^1 (i, j) & \parallel &
%			   \proc{Q}^1 (j, i) & \parallel &
%   \proc{T} \left( \lbag \rbag \right) \\
%\proc{C}^{34} (i, j) & = & \proc{P}^2 (i) & \parallel &
%			   \proc{Q}^1 (j, i) & \parallel &
%   \proc{T} \left( \lbag Q_j.c.P_{i + 1} \rbag \right) \\
%\proc{C}^{35} (i, j) & = & \proc{P}^2 (i) & \parallel &
%			   \proc{Q}^4 (j, i) & \parallel &
%   \proc{T} \left( \lbag \rbag \right) \\
%\\
%\proc{C}^{40} (i, j) & = & \proc{P}^2 (i - 1) & \parallel &
%			   \proc{Q}^2 (j) & \parallel &
%   \proc{T} \left( \lbag P_i.c.Q_{j + 1}, P_{i - 1}.c.Q_j \rbag \right) \\
%\proc{C}^{41} (i, j) & = & \proc{P}^5 (i - 1, j - 1) & \parallel &
%			   \proc{Q}^2 (j) & \parallel &
%   \proc{T} \left( \lbag P_i.c.Q_{j + 1} \rbag \right) \\
%\proc{C}^{42} (i, j) & = & \proc{P}^0 (i) & \parallel &
%			   \proc{Q}^2 (j) & \parallel &
%   \proc{T} \left( \lbag P_i.c.Q_{j + 1}, P_i.r.Q_j \rbag \right) \\
%\proc{C}^{43} (i, j) & = & \proc{P}^1 (i, j) & \parallel &
%			   \proc{Q}^2 (j) & \parallel &
%   \proc{T} \left( \lbag P_i.c.Q_{j + 1} \rbag \right) \\
%\\
%\proc{C}^{50} (i, j) & = & \proc{P}^6 (i - 1, j) & \parallel &
%			   \proc{Q}^2 (j) & \parallel &
%   \proc{T} \left( \lbag P_{i - 1}.c.Q_j \rbag \right) \\
%\proc{C}^{51} (i, j) & = & \proc{P}^7 (i - 1, j) & \parallel &
%			   \proc{Q}^2 (j) & \parallel &
%   \proc{T} \left( \lbag \rbag \right) \\
%\proc{C}^{52} (i, j) & = & \proc{P}^3 (i, j) & \parallel &
%			   \proc{Q}^2 (j) & \parallel &
%   \proc{T} \left( \lbag P_i.r.Q_j \rbag \right) \\
%\proc{C}^{53} (i, j) & = & \proc{P}^4 (i, j) & \parallel &
%			   \proc{Q}^2 (j) & \parallel &
%   \proc{T} \left( \lbag \rbag \right) \\
%\end{array}
%\end{displaymath}

\begin{displaymath}
\begin{array}{lclclcl}
\proc{C}^{00} (i, j) & = & \proc{P}^0 (i) & \parallel &
			   \proc{Q}^0 (j) & \parallel &
   \proc{T} \left( \lbag Q_{j - 1}.c.P_i, P_{i - 1}.c.Q_j \rbag \right) \\
\proc{C}^{01} (i, j) & = & \proc{P}^1 (i, j) & \parallel &
			   \proc{Q}^0 (j) & \parallel &
   \proc{T} \left( \lbag Q_{j - 1}.c.P_i \rbag \right) \\
\proc{C}^{02} (i, j) & = & \proc{P}^2 (i) & \parallel &
			   \proc{Q}^0 (j) & \parallel &
   \proc{T} \left( \lbag Q_{j - 1}.c.P_i, P_i.r.Q_j \rbag \right) \\
\proc{C}^{03} (i, j) & = & \proc{P}^3 (i, j) & \parallel &
			   \proc{Q}^0 (j) & \parallel &
   \proc{T} \left( \lbag Q_{j - 1}.c.P_i \rbag \right) \\
\proc{C}^{04} (i, j) & = & \proc{P}^0 (i + 1) & \parallel &
			   \proc{Q}^0 (j) & \parallel &
   \proc{T} \left( \lbag Q_{j - 1}.c.P_i, Q_j.c.P_{i + 1} \rbag \right) \\
\proc{C}^{05} (i, j) & = & \proc{P}^0 (i + 1) & \parallel &
			   \proc{Q}^4 (j, i) & \parallel &
   \proc{T} \left( \lbag Q_{j - 1}.c.P_i \rbag \right) \\
\\
\proc{C}^{10} (i, j) & = & \proc{P}^0 (i) & \parallel &
			   \proc{Q}^1 (j, i) & \parallel &
   \proc{T} \left( \lbag P_{i - 1}.c.Q_j \rbag \right) \\
\proc{C}^{11} (i, j) & = & \proc{P}^1 (i, j) & \parallel &
			   \proc{Q}^1 (j, i) & \parallel &
   \proc{T} \left( \lbag \rbag \right) \\
\proc{C}^{12} (i, j) & = & \proc{P}^2 (i) & \parallel &
			   \proc{Q}^1 (j, i) & \parallel &
   \proc{T} \left( \lbag P_i.r.Q_j \rbag \right) \\
\proc{C}^{13} (i, j) & = & \proc{P}^3 (i, j) & \parallel &
			   \proc{Q}^1 (j, i) & \parallel &
   \proc{T} \left( \lbag \rbag \right) \\
\proc{C}^{14} (i, j) & = & \proc{P}^0 (i + 1) & \parallel &
			   \proc{Q}^1 (j, i) & \parallel &
   \proc{T} \left( \lbag Q_j.c.P_{i + 1} \rbag \right) \\
\proc{C}^{15} (i, j) & = & \proc{P}^0 (i + 1) & \parallel &
			   \proc{Q}^5 (j, i) & \parallel &
   \proc{T} \left( \lbag \rbag \right) \\
\\
\proc{C}^{20} (i, j) & = & \proc{P}^0 (i) & \parallel &
			   \proc{Q}^2 (j) & \parallel &
   \proc{T} \left( \lbag Q_j.r.P_i, P_{i - 1}.c.Q_j \rbag \right) \\
\proc{C}^{21} (i, j) & = & \proc{P}^1 (i, j) & \parallel &
			   \proc{Q}^2 (j) & \parallel &
   \proc{T} \left( \lbag Q_j.r.P_i \rbag \right) \\
\proc{C}^{22} (i, j) & = & \proc{P}^2 (i) & \parallel &
			   \proc{Q}^2 (j) & \parallel &
   \proc{T} \left( \lbag Q_j.r.P_i, P_i.r.Q_j \rbag \right) \\
\proc{C}^{23} (i, j) & = & \proc{P}^3 (i, j) & \parallel &
			   \proc{Q}^2 (j) & \parallel &
   \proc{T} \left( \lbag Q_j.r.P_i \rbag \right) \\
\proc{C}^{24} (i, j) & = & \proc{P}^0 (i + 1) & \parallel &
			   \proc{Q}^2 (j) & \parallel &
   \proc{T} \left( \lbag Q_j.r.P_i, Q_j.c.P_{i + 1} \rbag \right) \\
\proc{C}^{25} (i, j) & = & \proc{P}^0 (i + 1) & \parallel &
			   \proc{Q}^6 (j, i) & \parallel &
   \proc{T} \left( \lbag Q_j.r.P_i \rbag \right) \\
\\
\proc{C}^{30} (i, j) & = & \proc{P}^0 (i) & \parallel &
			   \proc{Q}^3 (j, i) & \parallel &
   \proc{T} \left( \lbag P_{i - 1}.c.Q_j \rbag \right) \\
\proc{C}^{31} (i, j) & = & \proc{P}^1 (i, j) & \parallel &
			   \proc{Q}^3 (j, i) & \parallel &
   \proc{T} \left( \lbag \rbag \right) \\
\proc{C}^{32} (i, j) & = & \proc{P}^2 (i) & \parallel &
			   \proc{Q}^3 (j, i) & \parallel &
   \proc{T} \left( \lbag P_i.r.Q_j \rbag \right) \\
\proc{C}^{33} (i, j) & = & \proc{P}^3 (i, j) & \parallel &
			   \proc{Q}^3 (j, i) & \parallel &
   \proc{T} \left( \lbag \rbag \right) \\
\proc{C}^{34} (i, j) & = & \proc{P}^0 (i + 1) & \parallel &
			   \proc{Q}^3 (j, i) & \parallel &
   \proc{T} \left( \lbag Q_j.c.P_{i + 1} \rbag \right) \\
\proc{C}^{35} (i, j) & = & \proc{P}^0 (i + 1) & \parallel &
			   \proc{Q}^7 (j, i) & \parallel &
   \proc{T} \left( \lbag \rbag \right) \\
\\
\proc{C}^{40} (i, j) & = & \proc{P}^0 (i) & \parallel &
			   \proc{Q}^0 (j + 1) & \parallel &
   \proc{T} \left( \lbag P_i.c.Q_{j + 1}, P_{i - 1}.c.Q_j \rbag \right) \\
\proc{C}^{41} (i, j) & = & \proc{P}^1 (i, j) & \parallel &
			   \proc{Q}^0 (j + 1) & \parallel &
   \proc{T} \left( \lbag P_i.c.Q_{j + 1} \rbag \right) \\
\proc{C}^{42} (i, j) & = & \proc{P}^2 (i) & \parallel &
			   \proc{Q}^0 (j + 1) & \parallel &
   \proc{T} \left( \lbag P_i.c.Q_{j + 1}, P_i.r.Q_j \rbag \right) \\
\proc{C}^{43} (i, j) & = & \proc{P}^3 (i, j) & \parallel &
			   \proc{Q}^0 (j + 1) & \parallel &
   \proc{T} \left( \lbag P_i.c.Q_{j + 1} \rbag \right) \\
\\
\proc{C}^{50} (i, j) & = & \proc{P}^4 (i, j) & \parallel &
			   \proc{Q}^0 (j + 1) & \parallel &
   \proc{T} \left( \lbag P_{i - 1}.c.Q_j \rbag \right) \\
\proc{C}^{51} (i, j) & = & \proc{P}^5 (i, j) & \parallel &
			   \proc{Q}^0 (j + 1) & \parallel &
   \proc{T} \left( \lbag \rbag \right) \\
\proc{C}^{52} (i, j) & = & \proc{P}^6 (i, j) & \parallel &
			   \proc{Q}^0 (j + 1) & \parallel &
   \proc{T} \left( \lbag P_i.r.Q_j \rbag \right) \\
\proc{C}^{53} (i, j) & = & \proc{P}^7 (i, j) & \parallel &
			   \proc{Q}^0 (j + 1) & \parallel &
   \proc{T} \left( \lbag \rbag \right) \\
\end{array}
\end{displaymath}

\end{document}

