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\usepackage{tikz}
\usetikzlibrary{automata}

\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions

\begin{document}

\section*{Homework 7}

\begin{enumerate}
\item Suppose the following finite deterministic automaton over the alphabet $\{0, 1\}$.

\begin{center}
\begin{tikzpicture}[scale=2, line width=0.5mm]
  \node[state, initial, accepting]        (q0) at ( 0,1) {$q_0$};
  \node[state, accepting]                    (q1) at ( 1,1) {$q_1$};
 \node[state] (q2) at ( 2,1) {$q_2$};
  \path[->] (q0) edge[bend left] node[above] {$0$} (q1)
                  (q1) edge[bend left] node[above] {$1$} (q0)
                  (q2) edge[bend left=50] node[below] {$1$} (q0)
                  (q1) edge node[above] {$0$} (q2)
                  (q2) edge [loop right] node {$0$} ()
                  (q0) edge [loop below] node {$1$} ()
            ;
\end{tikzpicture}
\end{center}

Give a regular expression that can recognise the same language as
this automaton. (Hint: If you use Brzozwski's method, you can assume
Arden's lemma which states that an equation of the form $q = q\cdot r + s$
has the unique solution $q = s \cdot r^*$.)

\item Consider the following grammar 

\begin{center}
\begin{tabular}{l}
$S \rightarrow N\cdot P$\\
$P \rightarrow V\cdot N$\\
$N \rightarrow N\cdot N$\\
$N \rightarrow A \cdot N$\\
$N \rightarrow \texttt{student} \;|\; \texttt{trainer} \;|\; \texttt{team} \;|\; \texttt{trains}$\\
$V \rightarrow \texttt{trains} \;|\; \texttt{team}$\\
$A \rightarrow \texttt{The} \;|\; \texttt{the}$\\
\end{tabular}
\end{center}

where $S$ is the start symbol and $S$, $P$, $N$, $V$ and $A$ are non-terminals.
Using the CYK-algorithm, check whether or not the following string can be parsed
by the grammar:

\begin{center}
\texttt{The trainer trains the student team}
\end{center}

\item {\bf (Optional)} The task is to match strings where the letters are in alphabetical order---for example, 
\texttt{abcfjz} would pass, but \texttt{acb} would not. Whitespace should be ignored---for example
\texttt{ab c d} should pass. The point is to try to get the regular expression as short as possible!
See:

\begin{center}
\url{http://callumacrae.github.com/regex-tuesday/challenge11.html}
\end{center}
\end{enumerate}

\end{document}

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