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\section*{Homework 4}

\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 How many basic regular expressions are there to match the string

  $abcd$? (ii) How many if they cannot include $\epsilon$ and

  $\varnothing$? (iii) How many if they are also not allowed to

  contain stars? (iv) How many if they are also not allowed to contain

  $\_ + \_$?

%\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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