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% explain what is a context-free grammar and the language it generates 

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\section*{Homework 5}

\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}

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