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\documentclass{article}
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\usepackage{charter}
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\usepackage{hyperref}
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\usepackage{amssymb}
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\usepackage{amsmath}
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\usepackage{tikz}
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\usetikzlibrary{automata}
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\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
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\begin{document}
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\section*{Homework 7}
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\begin{enumerate}
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\item Suppose the following finite deterministic automaton over the alphabet $\{0, 1\}$.
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\begin{center}
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\begin{tikzpicture}[scale=2, line width=0.5mm]
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\node[state, initial, accepting] (q0) at ( 0,1) {$q_0$};
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\node[state, accepting] (q1) at ( 1,1) {$q_1$};
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\node[state] (q2) at ( 2,1) {$q_2$};
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\path[->] (q0) edge[bend left] node[above] {$0$} (q1)
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(q1) edge[bend left] node[above] {$1$} (q0)
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(q2) edge[bend left=50] node[below] {$1$} (q0)
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(q1) edge node[above] {$0$} (q2)
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(q2) edge [loop right] node {$0$} ()
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(q0) edge [loop below] node {$1$} ()
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;
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\end{tikzpicture}
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\end{center}
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Give a regular expression that can recognise the same language as
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this automaton. (Hint: If you use Brzozwski's method, you can assume
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Arden's lemma which states that an equation of the form $q = q\cdot r + s$
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has the unique solution $q = s \cdot r^*$.)
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\item Consider the following grammar
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\begin{center}
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\begin{tabular}{l}
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$S \rightarrow N\cdot P$\\
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$P \rightarrow V\cdot N$\\
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$N \rightarrow N\cdot N$\\
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$N \rightarrow A \cdot N$\\
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$N \rightarrow \texttt{student} \;|\; \texttt{trainer} \;|\; \texttt{team} \;|\; \texttt{trains}$\\
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$V \rightarrow \texttt{trains} \;|\; \texttt{team}$\\
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$A \rightarrow \texttt{The} \;|\; \texttt{the}$\\
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\end{tabular}
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\end{center}
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where $S$ is the start symbol and $S$, $P$, $N$, $V$ and $A$ are non-terminals.
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Using the CYK-algorithm, check whether or not the following string can be parsed
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by the grammar:
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\begin{center}
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\texttt{The trainer trains the student team}
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\end{center}
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\item {\bf (Optional)} The task is to match strings where the letters are in alphabetical order---for example,
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\texttt{abcfjz} would pass, but \texttt{acb} would not. Whitespace should be ignored---for example
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\texttt{ab c d} should pass. The point is to try to get the regular expression as short as possible!
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See:
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\begin{center}
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\url{http://callumacrae.github.com/regex-tuesday/challenge11.html}
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\end{center}
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\end{enumerate}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: t
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%%% End:
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