\documentclass{article}\usepackage{../style}\usepackage{../grammar}\begin{document}\section*{Homework 7}\HEADER\begin{enumerate}\item Suppose the context-sensitive grammar\begin{center}\begin{tabular}{lcl}$S$ & $::=$ & $bSAA\;|\; \epsilon$\\$A$ & $::=$ & $a$\\$bA$ & $::=$ & $Ab$\\\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 thestrings recognised by this grammar.\item Consider the following grammar \begin{plstx}[margin=1cm] : \meta{S\/} ::= \meta{N\/}\cdot \meta{P\/}\\ : \meta{P\/} ::= \meta{V\/}\cdot \meta{N\/}\\ : \meta{N\/} ::= \meta{N\/}\cdot \meta{N\/}\\ : \meta{N\/} ::= \meta{A\/}\cdot \meta{N\/}\\ : \meta{N\/} ::= \texttt{student} \mid \texttt{trainer} \mid \texttt{team} \mid \texttt{trains}\\ : \meta{V\/} ::= \texttt{trains} \mid \texttt{team}\\ : \meta{A\/} ::= \texttt{The} \mid \texttt{the}\\\end{plstx}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 parsedby the grammar:\begin{center}\texttt{The trainer trains the student team}\end{center}\item Transform the grammar\begin{center}\begin{tabular}{lcl}$A$ & $::=$ & $0A1 \;|\; BB$\\$B$ & $::=$ & $\epsilon \;|\; 2B$\end{tabular}\end{center}\noindentinto Chomsky normal form.\item Consider the following grammar $G$\begin{center}\begin{tabular}{l}$S ::= \texttt{if0} \cdot E \cdot \texttt{then} \cdot S$\\$S ::= \texttt{print} \cdot S$\\$S ::= \texttt{begin} \cdot B\cdot \texttt{end}$\\$B ::= S\cdot \texttt{;}$\\$B ::= S\cdot \texttt{;} \cdot B$\\$S ::= num$\\$E ::= num$\\$B ::= num$\end{tabular}\end{center}where $S$ is the start symbol and $S$, $E$ and $B$ arenon-terminals.Check each rule below and decide whether, when added to $G$,the combined grammar is ambiguous. If yes, give a string thathas more than one parse tree.\begin{center}\begin{tabular}{rl}(i) & $S ::= \texttt{if0} \cdot E\cdot \texttt{then} \cdot S\cdot \texttt{else} \cdot S$\\(ii) & $B ::= B \cdot B$\\(iii) & $E ::= ( \cdot E \cdot )$\\(iv) & $E ::= E \cdot + \cdot E$\end{tabular}\end{center}\item Suppose the string $``9-5+2''$. Give all ASTs that the following two grammars generate for this string.Grammar 1, where List is the starting symbol:\begin{center}\begin{tabular}{lcl}$List$ & $::=$ & $List + Digit \mid List - Digit \mid Digit$\\$Digit$ & $::=$ & $0 \mid 1 \mid 2 \mid 3 \mid 4 \mid 5 \mid 6 \mid 7 \mid 8 \mid 9$\end{tabular}\end{center}Grammar 2, where String is the starting symbol:\begin{center}\begin{tabular}{@{}lcl@{}} $String$ & $::=$ & $String + String \mid String - String \mid$\\ & & $0 \mid 1 \mid 2 \mid 3 \mid 4 \mid 5 \mid 6 \mid 7 \mid 8 \mid 9$\end{tabular}\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: