--- a/hws/hw05.tex	Sat Nov 01 15:06:41 2014 +0000
+++ b/hws/hw05.tex	Sat Nov 01 16:19:05 2014 +0000
@@ -12,6 +12,42 @@
 \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}
@@ -24,15 +60,32 @@
 \end{tabular}
 \end{center}
 
-in terms of the usual regular expressions
+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?
 
-\item Recall the definitions for $Der$ and $der$ from the lectures. 
+\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 
 
 \[
@@ -40,6 +93,7 @@
 \]
 
 holds.
+
 \end{enumerate}
 
 \end{document}