This coursework is worth 5\% and is due on 13 October at 16:00. You

are asked to implement a regular expression matcher and submit a

document containing the answers for the questions below. You can do

the implementation in any programming language you like, but you need

to submit the source code with which you answered the

questions. However, the coursework will \emph{only} be judged

according to the answers. You can submit your answers in a txt-file or

pdf.

\subsubsection*{Disclaimer}

It should be understood that the work submitted represents your own effort. 

You have not copied from anyone else. An exception is the Scala code I

showed during the lectures, which you can use.\bigskip

\subsubsection*{Tasks}

The task is to implement a regular expression matcher based on

derivatives. The implementation should be able to deal with the usual

(basic) regular expressions

but also with the following extended regular expressions:

$L([c_1 c_2 \ldots c_n])$ & $\dn$ & $\{[c_1], [c_2], \ldots, [c_n]\}$\\ 

$L(r^+)$                  & $\dn$ & $\bigcup_{1\le i}. L(r)^i$\\

$L(r^?)$                  & $\dn$ & $L(r) \cup \{[]\}$\\

$L(r^{\{n,m\}})$           & $\dn$ & $\bigcup_{n\le i \le m}. L(r)^i$\\

$L(\sim{}r)$              & $\dn$ & $\mathbb{A} - L(r)$

whereby in the last clause the set $\mathbb{A}$ stands for the set of

\emph{all} strings.  So $\sim{}r$ means `all the strings that $r$

cannot match'. We assume ranges like $[a\mbox{-}z0\mbox{-}9]$ are a

shorthand for the regular expression

Be careful that your implementation of $nullable$ and $der$ satisfies

for every $r$ the following two properties:

\item $nullable(r)$ if and only if $[]\in L(r)$

\noindent

{\bf Important!} Your implementation should have explicit cases for the

basic regular expressions, but also for the extended regular expressions.

That means do not treat the extended regular expressions by just translating 

them into the basic ones. See also Question 2, where you asked to give

the rules for \textit{nullable} and \textit{der}.

What is your King's email address (you will need it in Question 2)?

\subsection*{Question 2 (marked with 2\%)}

This question does not require any implementation. From the lectures

you have seen the definitions for the functions \textit{nullable} and

\textit{der}.  Give the rules for the extended regular expressions:

\begin{center}

\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}

$nullable([c_1 c_2 \ldots c_n])$  & $\dn$ & $?$\\

$nullable(r^+)$                   & $\dn$ & $?$\\

$nullable(r^?)$                   & $\dn$ & $?$\\

$nullable(r^{\{n,m\}})$            & $\dn$ & $?$\\

$nullable(\sim{}r)$               & $\dn$ & $?$\medskip\\

$der c ([c_1 c_2 \ldots c_n])$  & $\dn$ & $?$\\

$der c (r^+)$                   & $\dn$ & $?$\\

$der c (r^?)$                   & $\dn$ & $?$\\

$der c (r^{\{n,m\}})$            & $\dn$ & $?$\\

$der c (\sim{}r)$               & $\dn$ & $?$\\

\end{tabular}

\end{center}

\subsection*{Question 3 (marked with 1\%)}

and calculate the derivative according to your email address. When

calculating the derivative, simplify all regular expressions as much

as possible, but at least apply the following six simplification

rules:

Write down your simplified derivative in the ``mathematicical''

notation using parentheses where necessary.

\subsection*{Question 4 (marked with 1\%)}

Consider the regular expression $/ \cdot * \cdot

(\sim{}([a\mbox{-}z]^* \cdot * \cdot / \cdot [a\mbox{-}z]^*)) \cdot *

\cdot /$ and decide wether the following four strings are matched by

this regular expression. Answer yes or no.

\subsection*{Question 5 (marked with 1\%)}

Let $r_1$ be the regular expression $a\cdot a\cdot a$ and $r_2$ be

$(a^{\{19,19\}}) \cdot (a^?)$.  Decide whether the following three

strings consisting of $a$s only can be matched by $(r_1^+)^+$.

Similarly test them with $(r_2^+)^+$. Again answer in all six cases

with yes or no. \medskip

These are strings are meant to be entirely made up of $a$s. Be careful

when copy-and-pasting the strings so as to not forgetting any $a$ and

to not introducing any other character.