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\section*{Homework 6}

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\begin{enumerate}
\item (i) Give the regular expressions for lexing a language
      consisting of whitespaces, identifiers (some letters
      followed by digits), numbers, operations \texttt{=},
      \texttt{<} and \texttt{>}, and the keywords \texttt{if},
      \texttt{then} and \texttt{else}. (ii) Decide whether the
      following strings can be lexed in this language?

\begin{enumerate}
\item \texttt{"if y4 = 3 then 1 else 3"}
\item \texttt{"if33 ifif then then23 else else 32"}
\item \texttt{"if x4x < 33 then 1 else 3"}
\end{enumerate}

In case they can, give the corresponding token sequences. (Hint: 
Observe the maximal munch rule and priorities of your regular
expressions that make the process of lexing unambiguous.)

\item Suppose the grammar

\begin{center}
\begin{tabular}{lcl}
$E$ & $\rightarrow$ &  $F \;|\; F \cdot * \cdot F \;|\; F \cdot \backslash \cdot F$\\
$F$ & $\rightarrow$ & $T \;|\; T \cdot \texttt{+} \cdot T \;|\; T \cdot \texttt{-} \cdot T$\\
$T$ & $\rightarrow$ & $num \;|\; \texttt{(} \cdot E \cdot \texttt{)}$\\
\end{tabular}
\end{center}

where $E$, $F$ and $T$ are non-terminals, $E$ is the starting symbol of the grammar, and $num$ stands for
a number token. Give a parse tree for the string \texttt{(3+3)+(2*3)}.

\item Define what it means for a grammar to be ambiguous. Give an example of
an ambiguous grammar.

\item Suppose boolean expressions are built up from

\begin{center}
\begin{tabular}{ll}
1.) & tokens for \texttt{true} and \texttt{false},\\
2.) & the infix operations \texttt{$\wedge$} and \texttt{$\vee$},\\
3.) & the prefix operation $\neg$, and\\
4.) & can be enclosed in parentheses.
\end{tabular}
\end{center}

(i) Give a grammar that can recognise such boolean expressions
and (ii) give a sample string involving all rules given in 1.-4.~that 
can be parsed by this grammar.

\item Given the regular expressions

\begin{center}
\begin{tabular}{ll}    
  1) & $(ab + a)\cdot (\ONE + b)$\\
  2) & $(aa + a)^*$\\
\end{tabular}
\end{center}

there are several values for how these regular expressions can
recognise the string (for 1) $ab$ and (for 2) $aaa$. Give in each case
all the values and indicate which one is the POSIX value.
\item \POSTSCRIPT        
\end{enumerate}

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