\documentclass{article}+
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\usepackage{charter}+
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\usepackage{hyperref}+
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\usepackage{amssymb}+
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\usepackage{amsmath}+
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\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}%+
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\begin{document}+
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\section*{Coursework 1}+
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This coursework is worth 3\% and is due on 12 November at 16:00. You are asked to implement +
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a regular expression matcher and submit a document containing the answers for the questions +
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below. You can do the implementation in any programming language you like, but you need +
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to submit the source code with which you answered the questions. However, the coursework +
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will \emph{only} be judged according to the answers only. \bigskip+
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+
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\noindent+
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The task is to implement a regular expression matcher based on derivatives. The implementation +
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should be able to deal with the usual regular expressions+
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+
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\[+
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\varnothing, \epsilon, c, r_1 + r_2, r_1 \cdot r_2, r^*+
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\]+
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+
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\noindent+
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but also with+
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+
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\begin{center}+
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\begin{tabular}{ll}+
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$[c_1 c_2 \ldots c_n]$ & a range of characters\\+
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$r^+$ & one or more times $r$\\+
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$r^?$ & optional $r$\\+
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$r^{\{n,m\}}$ & at least $n$-times $r$ but no more than $m$-times\\+
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$\sim{}r$ & not-regular expression of $r$\\+
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\end{tabular}+
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\end{center}+
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+
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\noindent+
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In the case of $r^{\{n,m\}}$ we have the convention that $0 \le n \le m$.+
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The meaning of these regular expressions is+
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+
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\begin{center}+
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\begin{tabular}{r@{\hspace{2mm}}c@{\hspace{2mm}}l}+
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$L([c_1 c_2 \ldots c_n])$ & $\dn$ & $\{"c_1", "c_2", \ldots, "c_n"\}$\\ +
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$L(r^+)$            & $\dn$ & $\bigcup_{1\le i}. L(r)^i$\\+
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$L(r^?)$            & $\dn$ & $L(r) \cup \{""\}$\\+
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$L(r^{\{n,m\}})$ & $\dn$ & $\bigcup_{n\le i \le m}. L(r)^i$\\+
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$L(\sim{}r)$       & $\dn$ & $UNIV - L(r)$+
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\end{tabular}+
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\end{center}+
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+
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\noindent+
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whereby in the last clause the set $UNIV$ stands for the set of \emph{all} strings.+
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So $\sim{}r$ means `all the strings that $r$ cannot match'. We assume ranges +
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like $[a\mbox{-}z0\mbox{-}9]$ are a shorthand for the regular expression+
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+
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\[+
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[a b c d\ldots z 0 1\ldots 9]\;.+
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\]+
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+
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\noindent +
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Be careful that your implementations for $nullable$ and $der$ satisfies for every $r$ the following two+
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properties:+
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+
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\begin{itemize}+
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\item $nullable(r)$ if and only if $""\in L(r)$+
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\item $L(der\,c\,r)) = Der\,c\,(L(r))$+
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\end{itemize}+
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+
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\subsection*{Question 1 (unmarked)}+
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+
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What is your King's email address (you will need it in the questions below)?\bigskip +
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+
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\subsection*{Question 2 (marked with 1\%)}+
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+
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Implement the following regular expression for email addresses+
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+
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\[+
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([a\mbox{-}z0\mbox{-}9\_\!\_\,.-]^+)\cdot @\cdot ([a\mbox{-}z0\mbox{-}9\,.-]^+)\cdot .\cdot ([a\mbox{-}z\,.]^{\{2,6\}})+
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\]+
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+
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\noindent+
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and calculate the derivative according to your email address. When calculating+
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the derivative, simplify all regular expressions as much as possible, but at least apply the following +
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six rules:+
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+
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\begin{center}+
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\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}l}+
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$r \cdot \varnothing$ & $\mapsto$ & $\varnothing$\\ +
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$\varnothing \cdot r$ & $\mapsto$ & $\varnothing$\\ +
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$r \cdot \epsilon$ & $\mapsto$ & $r$\\ +
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$\epsilon \cdot r$ & $\mapsto$ & $r$\\ +
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$r + \varnothing$ & $\mapsto$ & $r$\\ +
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$\varnothing + r$ & $\mapsto$ & $r$\\ +
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\end{tabular}+
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\end{center}+
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+
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\noindent+
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Write down your simplified derivative.+
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+
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\subsection*{Question 3 (marked with 1\%)}+
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+
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Consider the regular expression $/ \cdot * \cdot (\sim{}([a\mbox{-}z]^* \cdot * \cdot / \cdot [a\mbox{-}z]^*)) \cdot * \cdot /$ and decide+
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wether the following four strings are matched by this regular expression. Answer yes or no.+
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+
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\begin{enumerate}+
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\item "/**/"+
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\item "/*foobar*/"+
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\item "/*test*/test*/"+
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\item "/*test/*test*/"+
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\end{enumerate}+
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+
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\subsection*{Question 4 (marked with 1\%)}+
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+
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Let $r_1$ be the regular expression $a\cdot a\cdot a$ and $r_2$ be $(a^{\{19,19\}}) \cdot (a^?)$.+
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Decide whether the following three strings can be matched by $(r_1^+)^+$. Similarly test them with $(r_2^+)^+$. +
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Again answer in all six cases with yes or no.+
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+
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\begin{enumerate}+
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\item $"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\+
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aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\+
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aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"$+
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\item $"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\ +
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aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\ +
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aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"$+
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\item$"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\ +
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aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\\ +
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aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"$+
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\end{enumerate}+
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+
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+
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\end{document}+
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+
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