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\section*{Homework 6}

\HEADER

\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 Parsing combinators consist of atomic parsers, alternative
  parsers, sequence parsers and semantic actions.  What is the purpose
  of (1) atomic parsers and of (2) semantic actions?

\item Parser combinators can directly be given a string as
      input, without the need of a lexer. What are the
      advantages to first lex a string and then feed a
      sequence of tokens as input to the parser?

\item The injection function for sequence regular expressions is defined
      by three clauses:

\begin{center}
\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}
  $\inj\,(r_1 \cdot r_2)\,c\,\,Seq(v_1,v_2)$ & $\dn$  & $Seq(\inj\,r_1\,c\,v_1,v_2)$\\
  $\inj\,(r_1 \cdot r_2)\,c\,\,\Left(Seq(v_1,v_2))$ & $\dn$  & $Seq(\inj\,r_1\,c\,v_1,v_2)$\\
  $\inj\,(r_1 \cdot r_2)\,c\,\,Right(v)$ & $\dn$  & $Seq(\textit{mkeps}(r_1),\inj\,r_2\,c\,v)$\\
\end{tabular}
\end{center}

Explain why there are three cases in the injection function for sequence
regular expressions. 
      
\item \POSTSCRIPT        
\end{enumerate}

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