--- a/handouts/ho02.tex	Sat Oct 12 09:13:42 2013 +0100
+++ b/handouts/ho02.tex	Sat Oct 12 10:12:38 2013 +0100
@@ -57,11 +57,12 @@
 \]
 
 \noindent
-Clearly we cannot use the function $L$ directly in order to solve this problem, because in general
-the set of strings $L$ returns is infinite (recall what $L(a^*)$ is). In such cases there is no algorithm
-then can test exhaustively, whether a string is member of this set.
+we can look at an algorithm to solve this problem.
+Clearly we cannot use the function $L$ directly for this, because in general
+the set of strings $L$ returns is infinite (recall what $L(a^*)$ is). In such cases there is no way we can implement
+an exhaustive test for whether a string is member of this set or not.
 
-The algorithm we define below consists of two parts. One is the function $nullable$ which takes a
+The algorithm we will define below consists of two parts. One is the function $nullable$ which takes a
 regular expression as argument and decides whether it can match the empty string (this means it returns a 
 boolean). This can be easily defined recursively as follows:
 
@@ -84,11 +85,11 @@
 \]
 
 \noindent
-On the left-hand side we have a function we can implement; on the right we have its specification. 
+Note on the left-hand side we have a function we can implement; on the right we have its specification. 
 
-The other function is calculating a \emph{derivative} of a regular expression. This is a function
+The other function of our matching algorithm calculates a \emph{derivative} of a regular expression. This is a function
 which will take a regular expression, say $r$, and a character, say $c$, as argument and return 
-a new regular expression. Beware that the intuition behind this function is not so easy to grasp on first
+a new regular expression. Be careful that the intuition behind this function is not so easy to grasp on first
 reading. Essentially this function solves the following problem: if $r$ can match a string of the form
 $c\!::\!s$, what does the regular expression look like that can match just $s$. The definition of this
 function is as follows:
@@ -133,7 +134,7 @@
 
 \noindent
 which essentially transforms a set of strings $A$ by filtering out all strings that do not start with $c$ and then
-strip off the $c$ from all the remaining strings. For example suppose $A = \{"f\!oo", "bar", "f\!rak"\}$ then
+strips off the $c$ from all the remaining strings. For example suppose $A = \{"f\!oo", "bar", "f\!rak"\}$ then
 \[
 Der\,f\,A = \{"oo", "rak"\}\quad,\quad
 Der\,b\,A = \{"ar"\}  \quad \text{and} \quad
@@ -150,10 +151,21 @@
 
 \noindent
 This property clarifies what regular expression $der$ calculates, namely take the set of strings
-that $r$ can match ($L(r)$), filter out all strings not starting with $c$ and strip off the $c$ from the
+that $r$ can match (that is $L(r)$), filter out all strings not starting with $c$ and strip off the $c$ from the
 remaining strings---this is exactly the language that $der\,c\,r$ can match.
 
-For our matching algorithm we need to lift the notion of derivatives from characters to strings. This can be
+If we want to find out whether the string $"abc"$ is matched by the regular expression $r$
+then we can iteratively apply $Der$ as follows
+
+\begin{enumerate}
+\item $Der\,a\,(L(r))$
+\item $Der\,b\,(Der\,a\,(L(r)))$
+\item $Der\,c\,(Der\,b\,(Der\,a\,(L(r))))$
+\end{enumerate}
+
+\noindent
+In the last step we need to test whether the empty string is in the set. Our matching algorithm will work similarly, 
+just using regular expression instead of sets. For this we need to lift the notion of derivatives from characters to strings. This can be
 done using the following function, taking a string and regular expression as input and a regular expression 
 as output.