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\documentclass{article}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{../style}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{../graphics}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{document}
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\section*{Homework 4}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\HEADER
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{enumerate}
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\item Given the regular expressions
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\begin{center}
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\begin{tabular}{ll}
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1) & $(ab + a)\cdot (\ONE + b)$\\
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2) & $(aa + a)^*$\\
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\end{tabular}
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\end{center}
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there are several values for how these regular expressions can
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recognise the strings (for 1) $ab$ and (for 2) $aaa$. Give in each case
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\emph{all} the values and indicate which one is the POSIX value.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item If a regular expression $r$ does not contain any occurrence of $\ZERO$,
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is it possible for $L(r)$ to be empty? Explain why, or give a proof.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Define the tokens and regular expressions for a language
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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consisting of numbers, left-parenthesis $($, right-parenthesis $)$,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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identifiers and the operations $+$, $-$ and $*$. Can the following
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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strings in this language be lexed?
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{itemize}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item $(a + 3) * b$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item $)()++ -33$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item $(a / 3) * 3$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{itemize}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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In case they can, can you give the corresponding token
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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sequences.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Assume $r$ is nullable. Show that
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\[ 1 + r + r\cdot r \;\equiv\; r\cdot r
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\]
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holds.
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\item \textbf{(Deleted)} Assume that $s^{-1}$ stands for the operation of reversing a
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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string $s$. Given the following \emph{reversing} function on regular
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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expressions
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$rev(\ZERO)$ & $\dn$ & $\ZERO$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$rev(\ONE)$ & $\dn$ & $\ONE$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$rev(c)$ & $\dn$ & $c$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$rev(r_1 + r_2)$ & $\dn$ & $rev(r_1) + rev(r_2)$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$rev(r_1 \cdot r_2)$ & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$rev(r^*)$ & $\dn$ & $rev(r)^*$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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and the set
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$Rev\,A \dn \{s^{-1} \;|\; s \in A\}$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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prove whether
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$L(rev(r)) = Rev (L(r))$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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holds.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Assume the delimiters for comments are
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\texttt{$\slash$*} and \texttt{*$\slash$}. Give a
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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regular expression that can recognise comments of the
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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form
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\texttt{$\slash$*~\ldots{}~*$\slash$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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where the three dots stand for arbitrary characters, but
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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not comment delimiters. (Hint: You can assume you are
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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already given a regular expression written \texttt{ALL},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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that can recognise any character, and a regular
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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expression \texttt{NOT} that recognises the complement
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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of a regular expression.)
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Simplify the regular expression
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\[
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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(\ZERO \cdot (b \cdot c)) +
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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((\ZERO \cdot c) + \ONE)
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Does simplification always preserve the meaning of a
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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regular expression?
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item The Sulzmann \& Lu algorithm contains the function
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$mkeps$ which answers how a regular expression can match
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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the empty string. What is the answer of $mkeps$ for the
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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regular expressions:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\[
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{array}{l}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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(\ZERO \cdot (b \cdot c)) +
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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((\ZERO \cdot c) + \ONE)\\
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(a + \ONE) \cdot (\ONE + \ONE)\\
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a^*
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{array}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item What is the purpose of the record regular expression in
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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the Sulzmann \& Lu algorithm?
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\item Recall the functions \textit{nullable} and
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\textit{zeroable}. Define recursive functions
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\textit{atmostempty} (for regular expressions that match no
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string or only the empty string), \textit{somechars} (for
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regular expressions that match some non-empty string),
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\textit{infinitestrings} (for regular expressions that can match
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infinitely many strings).
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\item There are two kinds of automata that are generate for
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regular expression matching---DFAs and NFAs. (1) Regular expression engines like
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the one in Python generate NFAs. Explain what is the problem with such
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NFAs and what is the reason why they use NFAs. (2) Regular expression
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engines like the one in Rust generate DFAs. Explain what is the
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problem with these regex engines and also what is the problem with $a^{\{1000\}}$
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in these engines.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\item (Optional) The tokenizer in \texttt{regexp3.scala} takes as
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%argument a string and a list of rules. The result is a list of tokens. Improve this tokenizer so
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%that it filters out all comments and whitespace from the result.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\item (Optional) Modify the tokenizer in \texttt{regexp2.scala} so that it
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%implements the \texttt{findAll} function. This function takes a regular
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%expressions and a string, and returns all substrings in this string that
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%match the regular expression.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item \POSTSCRIPT
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\end{enumerate}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: t
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%%% End:
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