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+\documentclass{article}
+\usepackage{style}
+%%\usepackage{../langs}
+
+\begin{document}
+
+\section*{Coursework 3}
+
+This coursework is worth XXX\% and is due on XXXX at
+16:00. You are asked to implement a regular expression matcher.
+
+
+
+
+\subsubsection*{Disclaimer}
+
+It should be understood that the work you submit represents
+your own effort. You have not copied from anyone else. An
+exception is the Scala code I showed during the lectures or
+uploaded to KEATS, which you can freely use.\bigskip
+
+
+\subsubsection*{Task}
+
+The task is to implement a regular expression matcher based on
+derivatives of regular expressions. The implementation should
+be able to deal with the usual (basic) regular expressions
+
+\begin{center}
+\begin{tabular}{lcll}
+ $r$ & $::=$ & $\ZERO$ & cannot match anything\\
+ & $|$ & $\ONE$ & can only match the empty string\\
+ & $|$ & $c$ & can match a character $c$\\
+ & $|$ & $r_1 + r_2$ & can match either with $r_1$ or with $r_2$\\
+ & $|$ & $r_1 \cdot r_2$ & can match first with $r_1$ and then with $r_2$\\
+ & $|$ & $r^*$ & can match zero or more times $r$\\
+ & $|$ & $r^{\{\uparrow n\}}$ & can match zero upto $n$ times $r$\\
+ & $|$ & $r^{\{n\}}$ & can match exactly $n$ times $r$\\
+\end{tabular}
+\end{center}
+
+\noindent
+Implement a function called \textit{nullable} by recursion over
+regular expressions:
+
+\begin{center}
+\begin{tabular}{lcl}
+$\textit{nullable}(\ZERO)$ & $\dn$ & $\textit{false}$\\
+$\textit{nullable}(\ONE)$ & $\dn$ & $\textit{true}$\\
+$\textit{nullable}(c)$ & $\dn$ & $\textit{false}$\\
+$\textit{nullable}(r_1 + r_2)$ & $\dn$ & $\textit{nullable}(r_1) \vee \textit{nullable}(r_2)$\\
+$\textit{nullable}(r_1 \cdot r_2)$ & $\dn$ & $\textit{nullable}(r_1) \wedge \textit{nullable}(r_2)$\\
+$\textit{nullable}(r^*)$ & $\dn$ & $\textit{true}$\\
+$\textit{nullable}(r^{\{\uparrow n\}})$ & $\dn$ & $\textit{true}$\\
+$\textit{nullable}(r^{\{n\}})$ & $\dn$ &
+ $\textit{if}\;n = 0\; \textit{then} \; \textit{true} \; \textit{else} \; \textit{nullable}(r)$\\
+\end{tabular}
+\end{center}
+
+\begin{center}
+\begin{tabular}{lcl}
+$\textit{der}\;c\;(\ZERO)$ & $\dn$ & $\ZERO$\\
+$\textit{der}\;c\;(\ONE)$ & $\dn$ & $\ZERO$\\
+$\textit{der}\;c\;(d)$ & $\dn$ & $\textit{if}\; c = d\;\textit{then} \;\ONE \; \textit{else} \;\ZERO$\\
+$\textit{der}\;c\;(r_1 + r_2)$ & $\dn$ & $(\textit{der}\;c\;r_1) + (\textit{der}\;c\;r_2)$\\
+$\textit{der}\;c\;(r_1 \cdot r_2)$ & $\dn$ & $\textit{if}\;\textit{nullable}(r_1)$\\
+ & & $\textit{then}\;((\textit{der}\;c\;r_1)\cdot r_2) + (\textit{der}\;c\;r_2)$\\
+ & & $\textit{else}\;(\textit{der}\;c\;r_1)\cdot r_2$\\
+$\textit{der}\;c\;(r^*)$ & $\dn$ & $(\textit{der}\;c\;r)\cdot (r^*)$\\
+$\textit{der}\;c\;(r^{\{\uparrow n\}})$ & $\dn$ & $\textit{if}\;n = 0\;\textit{then}\;\ZERO\;\text{else}\;
+ (\textit{der}\;c\;r)\cdot (r^{\{\uparrow n-1\}})$\\
+$\textit{der}\;c\;(r^{\{n\}})$ & $\dn$ &
+ $\textit{if}\;n = 0\; \textit{then} \; \ZERO\; \textit{else}\;$\\
+ & & $\textit{if} \;\textit{nullable}(r)\;\textit{then}\;(\textit{der}\;c\;r)\cdot (r^{\{\uparrow n-1\}})$\\
+ & & $\textit{else}\;(\textit{der}\;c\;r)\cdot (r^{\{n-1\}})$
+\end{tabular}
+\end{center}
+
+
+Be careful that your implementation of \textit{nullable} and
+\textit{der}\;c\; satisfies for every $r$ the following two
+properties (see also Question 2):
+
+\begin{itemize}
+\item $\textit{nullable}(r)$ if and only if $[]\in L(r)$
+\item $L(der\,c\,r) = Der\,c\,(L(r))$
+\end{itemize}
+
+\noindent {\bf Important!} Your implementation should have
+explicit cases for the basic regular expressions, but also
+explicit cases 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 are asked to explicitly give the rules for
+\textit{nullable} and \textit{der}\;c\; for the extended regular
+expressions.
+
+
+\subsection*{Question 1}
+
+What is your King's email address (you will need it in
+Question 3)?
+
+\subsection*{Question 2}
+
+This question does not require any implementation. From the
+lectures you have seen the definitions for the functions
+\textit{nullable} and \textit{der}\;c\; for the basic regular
+expressions. Give the rules for the extended regular
+expressions:
+
+\begin{center}
+\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
+$\textit{nullable}([c_1 c_2 \ldots c_n])$ & $\dn$ & $?$\\
+$\textit{nullable}(r^+)$ & $\dn$ & $?$\\
+$\textit{nullable}(r^?)$ & $\dn$ & $?$\\
+$\textit{nullable}(r^{\{n,m\}})$ & $\dn$ & $?$\\
+$\textit{nullable}(\sim{}r)$ & $\dn$ & $?$\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}
+
+\noindent
+Remember your definitions have to satisfy the two properties
+
+\begin{itemize}
+\item $\textit{nullable}(r)$ if and only if $[]\in L(r)$
+\item $L(der\,c\,r)) = Der\,c\,(L(r))$
+\end{itemize}
+
+\subsection*{Question 3}
+
+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 by applying the
+following 7 simplification rules:
+
+\begin{center}
+\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}ll}
+$r \cdot \ZERO$ & $\mapsto$ & $\ZERO$\\
+$\ZERO \cdot r$ & $\mapsto$ & $\ZERO$\\
+$r \cdot \ONE$ & $\mapsto$ & $r$\\
+$\ONE \cdot r$ & $\mapsto$ & $r$\\
+$r + \ZERO$ & $\mapsto$ & $r$\\
+$\ZERO + r$ & $\mapsto$ & $r$\\
+$r + r$ & $\mapsto$ & $r$\\
+\end{tabular}
+\end{center}
+
+\noindent Write down your simplified derivative in a readable
+notation using parentheses where necessary. That means you
+should use the infix notation $+$, $\cdot$, $^*$ and so on,
+instead of code.
+
+\subsection*{Question 4}
+
+Suppose \textit{[a-z]} stands for the range regular expression
+$[a,b,c,\ldots,z]$. 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 \texttt{"/**/"}
+\item \texttt{"/*foobar*/"}
+\item \texttt{"/*test*/test*/"}
+\item \texttt{"/*test/*test*/"}
+\end{enumerate}
+
+\noindent
+Also test your regular expression matcher with the regular
+expression $a^{\{3,5\}}$ and the strings
+
+\begin{enumerate}
+\setcounter{enumi}{4}
+\item \texttt{aa}
+\item \texttt{aaa}
+\item \texttt{aaaaa}
+\item \texttt{aaaaaa}
+\end{enumerate}
+
+\noindent
+Does your matcher produce the expected results?
+
+\subsection*{Question 5}
+
+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 \texttt{"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\
+aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\
+aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"}
+\item \texttt{"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\
+aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\
+aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"}
+\item \texttt{"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\
+aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\
+aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"}
+\end{enumerate}
+
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End: