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\section*{Homework 4}

\begin{enumerate}

\item Why is every finite set of strings a regular language?

\item What is the language recognised by the regular expressions $(\varnothing^*)^*$.

\item If a regular expression $r$ does not contain any occurrence of $\varnothing$ 

is it possible for $L(r)$ to be empty?

\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 Give a regular expression over the alphabet $\{a,b\}$ recognising all strings 

that do not contain any substring $bb$ and end in $a$.

\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 Given the alphabet $\{a,b\}$. Draw the automaton that has two states, say  $q_0$ and $q_1$.

The starting state is $q_0$ and the final state is $q_1$. The transition

function is given by

\begin{center}

\begin{tabular}{l}

$(q_0, a) \rightarrow q_0$\\

$(q_0, b) \rightarrow q_1$\\

$(q_1, b) \rightarrow q_1$

\end{tabular}

\end{center}

What is the languages recognised by this automaton?

\item Give a deterministic finite automaton that can recognise 

the language $L(a^*\cdot b\cdot b^*)$. 

\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}

% explain what is a context-free grammar and the language it generates 

%

%

% Define the language L(M) accepted by a deterministic finite automaton M.

%

%

% does (a + b)*b+ and (a*b+) + (b*b+) define the same language

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