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\section*{Homework 7}

\begin{enumerate}

\item Suppose the context-sensitive grammar

\begin{center}

\begin{tabular}{lcl}

$S$ & $\rightarrow$ &  $bSAA\;|\; \epsilon$\\

$A$ & $\rightarrow$ & $a$\\

$bA$ & $\rightarrow$ & $Ab$\\

\end{tabular}

\end{center}

where $S$ is the starting symbol of the grammar. 

Give a derivation  of the string $"\!aaabaaabb"$. 

What can you say about the number of as and bs in the

strings recognised by this grammar.

\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 Transform the grammar

\begin{center}

\begin{tabular}{lcl}

$A$ & $\rightarrow$ & $0A1 \;|\; BB$\\

$B$ & $\rightarrow$ & $\epsilon \;|\; 2B$

\end{tabular}

\end{center}

\noindent

into Chomsky normal form.

%\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}

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