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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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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{document}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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267
Christian Urban <christian dot urban at kcl dot ac dot uk>
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% explain what is a context-free grammar and the language it generates
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\section*{Homework 5}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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\begin{enumerate}
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294
Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Consider the basic 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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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$r ::= \varnothing \;|\; \epsilon \;|\; c \;|\; r_1 + r_2 \;|\; r_1 \cdot r_2 \;|\; r^*$
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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and suppose you want to show a property $P(r)$ for all regular
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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expressions $r$ by structural induction. Write down which cases do you
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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need to analyse. State clearly the induction hypotheses if applicable
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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in a case.
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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 Define a regular expression, written $ALL$, that can match
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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every string. This definition should be in terms of the
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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following extended 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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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$r ::= \varnothing \;|\;
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\epsilon \;|\;
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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c \;|\;
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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r_1 + r_2 \;|\;
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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r_1 \cdot r_2 \;|\;
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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r^* \;|\;
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\sim r$
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\item Assume the delimiters for comments are \texttt{$\slash$*}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%and \texttt{*$\slash$}. Give a regular expression that can
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%recognise comments of the form
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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{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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%
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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 not
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%comment delimiters.
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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>
parents:
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\item Define the following regular expressions
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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\begin{tabular}{ll}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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$r^+$ & (one or more matches)\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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$r^?$ & (zero or one match)\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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$r^{\{n\}}$ & (exactly $n$ matches)\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$r^{\{m, n\}}$ & (at least $m$ and maximal $n$ matches, with the\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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& \phantom{(}assumption $m \le n$)\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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in terms of the usual basic regular expressions
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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$r ::= \varnothing \;|\; \epsilon \;|\; c \;|\; r_1 + r_2 \;|\; r_1 \cdot r_2 \;|\; r^*$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item Give the regular expressions for lexing a language
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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consisting of identifiers, left-parenthesis \texttt{(},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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right-parenthesis \texttt{)}, numbers that can be either
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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positive or negative, and the operations \texttt{+},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\texttt{-} and \texttt{*}.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Decide whether the following strings
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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can be lexed in this language?
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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{enumerate}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item \texttt{"(a3+3)*b"}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item \texttt{")()++-33"}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item \texttt{"(b42/3)*3"}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{enumerate}
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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, give the corresponding token sequences. (Hint:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Observe the maximal munch rule and the priorities of your regular
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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expressions that make the process of lexing unambiguous.)
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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) Recall the definitions for $Der$ and $der$ from the lectures.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Prove by induction on $r$ the property that
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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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\[
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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L(der\,c\,r) = Der\,c\,(L(r))
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\]
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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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holds.
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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>
parents:
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\end{enumerate}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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\end{document}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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%%% Local Variables:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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%%% mode: latex
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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%%% TeX-master: t
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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%%% End:
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