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\section*{Coursework 1}

This coursework is worth 3\% and is due on 12 November 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.\bigskip

\noindent

The task is to implement a regular expression matcher based on derivatives. The implementation 

should be able to deal with the usual regular expressions

\[

\varnothing, \epsilon, c, r_1 + r_2, r_1 \cdot r_2, r^*

\]

\noindent

but also with

\begin{center}

\begin{tabular}{ll}

$[c_1 c_2 \ldots c_n]$ & a range of characters\\

$r^+$ & one or more times $r$\\

$r^?$ & optional $r$\\

$r^{\{n,m\}}$ & at least $n$-times $r$ but no more than $m$-times\\

$\sim{}r$ & not-regular expression of $r$\\

\end{tabular}

\end{center}

\noindent

In the case of $r^{\{n,m\}}$ we have the convention that $0 \le n \le m$.

The meaning of these regular expressions is

\begin{center}

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

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

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

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

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

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

\end{tabular}

\end{center}

\noindent

whereby in the last clause the set $UNIV$ 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

\[

[a b c d\ldots z 0 1\ldots 9]\;.

\]

\noindent 

Be careful that your implementation of $nullable$ and $der$ satisfies for every $r$ the following two

properties:

\begin{itemize}

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

\item $L(der\,c\,r)) = Der\,c\,(L(r))$

\end{itemize}

\newpage

\subsection*{Question 1 (unmarked)}

What is your King's email address (you will need it in the next question)?\bigskip 

\subsection*{Question 2 (marked with 1\%)}

Implement the following regular expression for email addresses

\[

([a\mbox{-}z0\mbox{-}9\_\!\_\,.-]^+)\cdot @\cdot ([a\mbox{-}z0\mbox{-}9\,.-]^+)\cdot .\cdot ([a\mbox{-}z\,.]^{\{2,6\}})

\]

\noindent

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:

\begin{center}

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

$r \cdot \varnothing$ & $\mapsto$ & $\varnothing$\\ 

$\varnothing \cdot r$ & $\mapsto$ & $\varnothing$\\ 

$r \cdot \epsilon$ & $\mapsto$ & $r$\\ 

$\epsilon \cdot r$ & $\mapsto$ & $r$\\ 

$r + \varnothing$ & $\mapsto$ & $r$\\ 

$\varnothing + r$ & $\mapsto$ & $r$\\ 

\end{tabular}

\end{center}

\noindent

Write down your simplified derivative in the ``mathematicical'' notation using parentheses where necessary.

\subsection*{Question 3 (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.

\begin{enumerate}

\item "/**/"

\item "/*foobar*/"

\item "/*test*/test*/"

\item "/*test/*test*/"

\end{enumerate}

\subsection*{Question 4 (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

\noindent

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.

\begin{enumerate}

\item $"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"$

\item $"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\ 

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\ 

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"$

\item$"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\ 

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\ 

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"$

\end{enumerate}

\end{document}

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