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+\documentclass{article}
+\usepackage{charter}
+\usepackage{hyperref}
+\usepackage{amssymb}
+\usepackage{amsmath}
+\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}
+
+%%% Local Variables: 
+%%% mode: latex
+%%% TeX-master: t
+%%% End: 




