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