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\documentclass{article}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{../style}
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\begin{document}
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\section*{Homework 7}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\HEADER
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{enumerate}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Suppose the context-sensitive grammar
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{tabular}{lcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$S$ & $\rightarrow$ & $bSAA\;|\; \epsilon$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$A$ & $\rightarrow$ & $a$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$bA$ & $\rightarrow$ & $Ab$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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where $S$ is the starting symbol of the grammar.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Give a derivation of the string $"\!aaabaaabb"$.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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What can you say about the number of as and bs in the
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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strings recognised by this grammar.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Transform the grammar
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{tabular}{lcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$A$ & $\rightarrow$ & $0A1 \;|\; BB$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$B$ & $\rightarrow$ & $\epsilon \;|\; 2B$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\noindent
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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into Chomsky normal form.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Consider the following grammar $G$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{tabular}{l}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$S \rightarrow \texttt{if0} \cdot E \cdot \texttt{then} \cdot S$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$S \rightarrow \texttt{print} \cdot S$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$S \rightarrow \texttt{begin} \cdot B\cdot \texttt{end}$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$B \rightarrow S\cdot \texttt{;}$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$B \rightarrow S\cdot \texttt{;} \cdot B$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$S \rightarrow num$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$E \rightarrow num$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$B \rightarrow num$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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where $S$ is the start symbol and $S$, $E$ and $B$ are
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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non-terminals.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Check each rule below and decide whether, when added to $G$,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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the combined grammar is ambiguous. If yes, give a string that
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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has more than one parse tree.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{tabular}{rl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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(i) & $S \rightarrow \texttt{if0} \cdot E\cdot \texttt{then} \cdot S\cdot \texttt{else} \cdot S$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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(ii) & $B \rightarrow B \cdot B$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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(iii) & $E \rightarrow ( \cdot E \cdot )$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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(iv) & $E \rightarrow E \cdot + \cdot E$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\texttt{abcfjz} would pass, but \texttt{acb} would not. Whitespace should be ignored---for example
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%See:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\url{http://callumacrae.github.com/regex-tuesday/challenge11.html}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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