\documentclass{article}\usepackage{../style}\usepackage{../langs}\usepackage{array}\begin{document}\newcolumntype{C}[1]{>{\centering}m{#1}}\section*{Coursework 1 (Strand 1)}This coursework is worth 4\% and is due on 19 October at16:00. You are asked to implement a regular expression matcherand submit a document containing the answers for the questionsbelow. You can do the implementation in any programminglanguage you like, but you need to submit the source code withwhich you answered the questions, otherwise a mark of 0\% willbe awarded. You can submit your answers in a txt-file or pdf.Code send as code.\subsubsection*{Disclaimer}It should be understood that the work you submit representsyour own effort. You have not copied from anyone else. Anexception is the Scala code I showed during the lectures oruploaded to KEATS, which you can freely use.\bigskip\noindentIf you have any questions, please send me an email in \textbf{good}time.\bigskip\subsection*{Task}The task is to implement a regular expression matcher based onderivatives of regular expressions. The implementation shouldbe able to deal with the usual (basic) regular expressions\[\ZERO,\; \ONE,\; c,\; r_1 + r_2,\; r_1 \cdot r_2,\; r^*\]\noindentbut also with the following extended regular expressions:\begin{center}\begin{tabular}{ll} $[c_1,c_2,\ldots,c_n]$ & a set of characters---for character ranges\\ $r^+$ & one or more times $r$\\ $r^?$ & optional $r$\\ $r^{\{n\}}$ & exactly $n$-times\\ $r^{\{..m\}}$ & zero or more times $r$ but no more than $m$-times\\ $r^{\{n..\}}$ & at least $n$-times $r$\\ $r^{\{n..m\}}$ & at least $n$-times $r$ but no more than $m$-times\\ $\sim{}r$ & not-regular-expression of $r$\\\end{tabular}\end{center}\noindent You can assume that $n$ and $m$ are greater or equal than$0$. In the case of $r^{\{n,m\}}$ you can also assume $0 \le n \le m$.\bigskip\noindent {\bf Important!} Your implementation should have explicitcases for the basic regular expressions, but also for explicit cases forthe extended regular expressions. That means do not treat the extendedregular expressions by just translating them into the basic ones. Seealso Question 2, where you are asked to explicitly give the rules for\textit{nullable} and \textit{der} for the extended regularexpressions.\newpage\noindentThe meanings of the extended regular expressions are\begin{center}\begin{tabular}{r@{\hspace{2mm}}c@{\hspace{2mm}}l} $L([c_1,c_2,\ldots,c_n])$ & $\dn$ & $\{[c_1], [c_2], \ldots, [c_n]\}$\\ $L(r^+)$ & $\dn$ & $\bigcup_{1\le i}.\;L(r)^i$\\ $L(r^?)$ & $\dn$ & $L(r) \cup \{[]\}$\\ $L(r^{\{n\}})$ & $\dn$ & $L(r)^n$\\ $L(r^{\{..m\}})$ & $\dn$ & $\bigcup_{0\le i \le m}.\;L(r)^i$\\ $L(r^{\{n..\}})$ & $\dn$ & $\bigcup_{n\le i}.\;L(r)^i$\\ $L(r^{\{n..m\}})$ & $\dn$ & $\bigcup_{n\le i \le m}.\;L(r)^i$\\ $L(\sim{}r)$ & $\dn$ & $\Sigma^* - L(r)$\end{tabular}\end{center}\noindent whereby in the last clause the set $\Sigma^*$ standsfor the set of \emph{all} strings over the alphabet $\Sigma$(in the implementation the alphabet can be just what isrepresented by, say, the type \pcode{Char}). So $\sim{}r$means in effect ``all the strings that $r$ cannot match''.\medskip \noindentBe careful that your implementation of \textit{nullable} and\textit{der} satisfies for every regular expression $r$ the followingtwo 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}\subsection*{Question 1 (Unmarked)}What is your King's email address (you will need it inQuestion 4)?\subsection*{Question 2}From thelectures you have seen the definitions for the functions\textit{nullable} and \textit{der} for the basic regularexpressions. Implement the rules for the extended regularexpressions:\begin{center}\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}} $\textit{nullable}([c_1,c_2,\ldots,c_n])$ & $\dn$ & $?$\\ $\textit{nullable}(r^+)$ & $\dn$ & $?$\\ $\textit{nullable}(r^?)$ & $\dn$ & $?$\\ $\textit{nullable}(r^{\{n\}})$ & $\dn$ & $?$\\ $\textit{nullable}(r^{\{..m\}})$ & $\dn$ & $?$\\ $\textit{nullable}(r^{\{n..\}})$ & $\dn$ & $?$\\ $\textit{nullable}(r^{\{n..m\}})$ & $\dn$ & $?$\\ $\textit{nullable}(\sim{}r)$ & $\dn$ & $?$\end{tabular}\end{center}\begin{center}\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}} $der\, c\, ([c_1,c_2,\ldots,c_n])$ & $\dn$ & $?$\\ $der\, c\, (r^+)$ & $\dn$ & $?$\\ $der\, c\, (r^?)$ & $\dn$ & $?$\\ $der\, c\, (r^{\{n\}})$ & $\dn$ & $?$\\ $der\, c\, (r^{\{..m\}})$ & $\dn$ & $?$\\ $der\, c\, (r^{\{n..\}})$ & $\dn$ & $?$\\ $der\, c\, (r^{\{n..m\}})$ & $\dn$ & $?$\\ $der\, c\, (\sim{}r)$ & $\dn$ & $?$\\\end{tabular}\end{center}\noindentRemember 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}\noindentGiven the definitions of \textit{nullable} and \textit{der}, it iseasy to implement a regular expression matcher. Test your regularexpression matcher with (at least) the examples:\begin{center}\def\arraystretch{1.2} \begin{tabular}{r|m{12mm}|m{12mm}|m{12mm}|m{12mm}|m{12mm}|m{12mm}} string & $a^{\{3\}}$ & $(a^?)^{\{3\}}$ & $a^{\{..3\}}$ & $(a^?)^{\{..3\}}$ & $a^{\{3..5\}}$ & $(a^?)^{\{3..5\}}$\\\hline $[]$ &&&&&& \\\hline \texttt{a} &&&&&& \\\hline \texttt{aa} &&&&&& \\\hline \texttt{aaa} &&&&&& \\\hline \texttt{aaaaa} &&&&&& \\\hline \texttt{aaaaaa}&&&&&& \\\end{tabular}\end{center}\noindentDoes your matcher produce the expected results?\subsection*{Question 3}As you can see, there are a number of explicit regular expressionsthat deal with single or several characters, for example:\begin{center}\begin{tabular}{ll} $c$ & matches a single character\\ $[c_1,c_2,\ldots,c_n]$ & matches a set of characters---for character ranges\\ $\textit{ALL}$ & matches any character\end{tabular}\end{center}\noindentthe latter is useful for matching any string (for exampleby using $\textit{ALL}^*$). In order to avoid having an explicit constructorfor each case, we can generalise all these cases and introduce a singleconstructor $\textit{CFUN}(f)$ where $f$ is a function from charactersto a boolean. The idea is that the function $f$ determines which character(s)are matched, namely those where $f$ returns \texttt{true}.In this question implement \textit{CFUN} and define\begin{center}\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}} $\textit{nullable}(\textit{CFUN}(f))$ & $\dn$ & $?$\\ $\textit{der}\,c\,(\textit{CFUN}(f))$ & $\dn$ & $?$\end{tabular}\end{center}\noindent in your matcher and then also give definitions for\begin{center}\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}} $c$ & $\dn$ & $\textit{CFUN}(?)$\\ $[c_1,c_2,\ldots,c_n]$ & $\dn$ & $\textit{CFUN}(?)$\\ $\textit{ALL}$ & $\dn$ & $\textit{CFUN}(?)$\end{tabular}\end{center}\subsection*{Question 4}Suppose $[a\mbox{-}z0\mbox{-}9\_\,.\mbox{-}]$ stands for the regular expression\[[a,b,c,\ldots,z,0,\dots,9,\_,.,\mbox{-}]\;.\]\noindentDefine in your code 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 emailaddress. When calculating the derivative, simplify all regularexpressions as much as possible by applying thefollowing 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 readablenotation using parentheses where necessary. That means youshould use the infix notation $+$, $\cdot$, $^*$ and so on,instead of code.\bigskip\noindentImplement the simplification rules in your regular expression matcher.Consider the regular expression $/ \cdot * \cdot(\sim{}(\textit{ALL}^* \cdot * \cdot / \cdot \textit{ALL}^*)) \cdot *\cdot /$ and decide wether the following four strings are matched bythis regular expression. Answer yes or no.\begin{enumerate}\item \texttt{"/**/"}\item \texttt{"/*foobar*/"}\item \texttt{"/*test*/test*/"}\item \texttt{"/*test/*test*/"}\end{enumerate}\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 threestrings consisting of $a$s only can be matched by $(r_1^+)^+$.Similarly test them with $(r_2^+)^+$. Again answer in all six caseswith yes or no. \medskip\noindentThese are strings are meant to be entirely made up of $a$s. Be carefulwhen copy-and-pasting the strings so as to not forgetting any $a$ andto not introducing any other character.\begin{enumerate}\setcounter{enumi}{4}\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: