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\section*{Homework 4}

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\begin{enumerate}

\item If a regular expression $r$ does not contain any occurrence of $\varnothing$,  
is it possible for $L(r)$ to be empty?

\item Define the tokens and regular expressions for a language
  consisting of numbers, left-parenthesis $($, right-parenthesis $)$,
  identifiers and the operations $+$, $-$ and $*$. Can the following
  strings in this language be lexed?

  \begin{itemize}
  \item $(a + 3) * b$
  \item $)()++ -33$
  \item $(a / 3) * 3$
  \end{itemize}

  In case they can, can you give the corresponding token
  sequences.

\item Assume that $s^{-1}$ stands for the operation of reversing a
  string $s$. Given the following \emph{reversing} function on regular
  expressions

  \begin{center}
    \begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}
      $rev(\varnothing)$   & $\dn$ & $\varnothing$\\
      $rev(\epsilon)$         & $\dn$ & $\epsilon$\\
      $rev(c)$                      & $\dn$ & $c$\\
      $rev(r_1 + r_2)$        & $\dn$ & $rev(r_1) + rev(r_2)$\\
      $rev(r_1 \cdot r_2)$  & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\
      $rev(r^*)$                   & $\dn$ & $rev(r)^*$\\
    \end{tabular}
  \end{center}

  and the set

  \begin{center}
    $Rev\,A \dn \{s^{-1} \;|\; s \in A\}$
  \end{center}

  prove whether

  \begin{center}
    $L(rev(r)) = Rev (L(r))$
  \end{center}

  holds.

\item Assume the delimiters for comments are \texttt{$\slash$*} and
  \texttt{*$\slash$}.  Give a regular expression that can recognise
  comments of the form

  \begin{center}
    \texttt{$\slash$*~\ldots{}~*$\slash$} 
  \end{center}

  where the three dots stand for arbitrary characters, but not comment delimiters.
  (Hint: You can assume you are already given a regular expression written \texttt{ALL},
  that can recognise any character, and a regular expression \texttt{NOT} that recognises
  the complement of a regular expression.)
  
\item Simplify the regular expression

\[
(\varnothing \cdot (b \cdot c)) + 
((\varnothing \cdot c) + \epsilon)
\]

Does simplification always preserve the meaning of a regular
expression? 
     
\item The Sulzmann algorithm contains the function $mkeps$
      which answers how a regular expression can match the
      empty string. What is the answer of $mkeps$ for the
      regular expressions:    

  \[
  \begin{array}{l}
  (\varnothing \cdot (b \cdot c)) + 
  ((\varnothing \cdot c) + \epsilon)\\
  (a + \varepsilon) \cdot (\varepsilon + \varepsilon)
  \end{array}
  \]

\item What is the purpose of the record regular expression
  in the Sulzmann algorithm?
  
  
%\item (Optional) The tokenizer in \texttt{regexp3.scala} takes as
%argument a string and a list of rules. The result is a list of tokens. Improve this tokenizer so 
%that it filters out all comments and whitespace from the result.

%\item (Optional) Modify the tokenizer in \texttt{regexp2.scala} so that it
%implements the \texttt{findAll} function. This function takes a regular
%expressions and a string, and returns all substrings in this string that 
%match the regular expression.
\end{enumerate}


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