--- a/cws/cw03.tex	Thu Nov 24 09:42:49 2016 +0000
+++ b/cws/cw03.tex	Thu Nov 24 10:41:42 2016 +0000
@@ -16,7 +16,7 @@
 
 \noindent
 \textbf{Important:} Do not use any mutable data structures in your
-submissions! They are not needed. This excluded the use of
+submission! They are not needed. This excluded the use of
 \texttt{ListBuffer}s, for example. Do not use \texttt{return} in your
 code! It has a different meaning in Scala, than in Java.  Do not use
 \texttt{var}! This declares a mutable variable.  Make sure the
@@ -34,10 +34,10 @@
 
 \subsection*{Part 1 (6 Marks)}
 
-The task is to implement a regular expression matcher based on
+The task is to implement a regular expression matcher that is based on
 derivatives of regular expressions. The implementation can deal
 with the following regular expressions, which have been predefined
-file re.scala:
+in the file re.scala:
 
 \begin{center}
 \begin{tabular}{lcll}
@@ -57,7 +57,7 @@
 expressions are one of the fastest and simplest ways to match patterns
 in text, and are endlessly useful for searching, editing and
 analysing text in all sorts of places. However, you need to be
-fast, otherwise you will stumble upon problems such as recently reported at
+fast, otherwise you will stumble over problems such as recently reported at
 
 {\small
 \begin{itemize}
@@ -70,7 +70,7 @@
 
 \begin{itemize}
 \item[(1a)] Implement a function, called \textit{nullable}, by recursion over
-  regular expressions. This function test whether a regular expression can match
+  regular expressions. This function tests whether a regular expression can match
   the empty string.
 
 \begin{center}
@@ -86,7 +86,8 @@
 
 \item[(1b)] Implement a function, called \textit{der}, by recursion over
   regular expressions. It takes a character and a regular expression
-  as arguments and calculates the derivative regular expression.
+  as arguments and calculates the derivative regular expression according
+  to the rules:
 
 \begin{center}
 \begin{tabular}{lcl}
@@ -99,16 +100,53 @@
       & & $\textit{else}\;(\textit{der}\;c\;r_1)\cdot r_2$\\
 $\textit{der}\;c\;(r^*)$ & $\dn$ & $(\textit{der}\;c\;r)\cdot (r^*)$\\
 \end{tabular}
-\end{center}\hfill[1 Mark]
+\end{center}
+
+For example given the regular expression $r = (a \cdot b) \cdot c$, the derivatives
+w.r.t.~the characters $a$, $b$ and $c$ are
+
+\begin{center}
+  \begin{tabular}{lcll}
+    $\textit{der}\;a\;r$ & $=$ & $(\ONE \cdot b)\cdot c$ & ($= r'$)\\
+    $\textit{der}\;b\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$\\
+    $\textit{der}\;c\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$
+  \end{tabular}
+\end{center}
+
+Let $r'$ stand for the first derivative, then taking the derivatives of $r'$
+w.r.t.~the characters $a$, $b$ and $c$ gives
+
+\begin{center}
+  \begin{tabular}{lcll}
+    $\textit{der}\;a\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$ \\
+    $\textit{der}\;b\;r'$ & $=$ & $((\ZERO \cdot b) + \ONE)\cdot c$ & ($= r''$)\\
+    $\textit{der}\;c\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$
+  \end{tabular}
+\end{center}
+
+One more example: Let $r''$ stand for the second derivative above,
+then taking the derivatives of $r''$ w.r.t.~the characters $a$, $b$
+and $c$ gives
+
+\begin{center}
+  \begin{tabular}{lcll}
+    $\textit{der}\;a\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$ \\
+    $\textit{der}\;b\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$\\
+    $\textit{der}\;c\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ONE$
+  \end{tabular}
+\end{center}
+
+Note, the last derivative can match the empty string, that is it is \textit{nullable}.\\
+\mbox{}\hfill\mbox{[1 Mark]}
 
 \item[(1c)] Implement the function \textit{simp}, which recursively
-  traverses a regular expression from inside to outside, and
-  simplifies every sub-regular-expressions on the left to
+  traverses a regular expression from the inside to the outside, and
+  simplifies every sub-regular-expression on the left (see below) to
   the regular expression on the right, except it does not simplify inside
   ${}^*$-regular expressions.
 
   \begin{center}
-\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}ll}
+\begin{tabular}{l@{\hspace{4mm}}c@{\hspace{4mm}}ll}
 $r \cdot \ZERO$ & $\mapsto$ & $\ZERO$\\ 
 $\ZERO \cdot r$ & $\mapsto$ & $\ZERO$\\ 
 $r \cdot \ONE$ & $\mapsto$ & $r$\\ 
@@ -119,43 +157,44 @@
 \end{tabular}
   \end{center}
 
-  For example
+  For example the regular expression
   \[(r_1 + \ZERO) \cdot \ONE + ((\ONE + r_2) + r_3) \cdot (r_4 \cdot \ZERO)\]
 
   simplifies to just $r_1$. 
   \hfill[1 Mark]
 
 \item[(1d)] Implement two functions: The first, called \textit{ders},
-  takes a list of characters as arguments and a regular expression and
-  buids the derivative as follows:
+  takes a list of characters and a regular expression as arguments, and
+  builds the derivative w.r.t.~the list as follows:
 
 \begin{center}
 \begin{tabular}{lcl}
-$\textit{ders}\;Nil\;(r)$ & $\dn$ & $r$\\
-  $\textit{ders}\;c::cs\;(r)$  & $\dn$ &
+$\textit{ders}\;(Nil)\;r$ & $\dn$ & $r$\\
+  $\textit{ders}\;(c::cs)\;r$  & $\dn$ &
     $\textit{ders}\;cs\;(\textit{simp}(\textit{der}\;c\;r))$\\
 \end{tabular}
 \end{center}
 
 The second, called \textit{matcher}, takes a string and a regular expression
 as arguments. It builds first the derivatives according to \textit{ders}
-and at the end tests whether the resulting redular expression can match
+and after that tests whether the resulting derivative regular expression can match
 the empty string (using \textit{nullable}).
-For example the \textit{matcher} will produce true if given the
-regular expression $a\cdot b\cdot c$ and the string $abc$.
+For example the \textit{matcher} will produce true given the
+regular expression $(a\cdot b)\cdot c$ and the string $abc$.
 \hfill[1 Mark]
 
-\item[(1e)] Implement the function \textit{replace}: it searches (from the left to 
-right) in string $s_1$ all the non-empty substrings that match the 
-regular expression---these substrings are assumed to be
+\item[(1e)] Implement the function $\textit{replace}\;r\;s_1\;s_2$: it searches
+  (from the left to 
+right) in the string $s_1$ all the non-empty substrings that match the 
+regular expression $r$---these substrings are assumed to be
 the longest substrings matched by the regular expression and
-assumed to be non-overlapping. All these substrings in $s_1$ are replaced
-by $s_2$. For example given the regular expression
+assumed to be non-overlapping. All these substrings in $s_1$ matched by $r$
+are replaced by $s_2$. For example given the regular expression
 
 \[(a \cdot a)^* + (b \cdot b)\]
 
-\noindent the string $aabbbaaaaaaabaaaaabbaaaabb$ and
-replacement string $c$ yields the string
+\noindent the string $s_1 = aabbbaaaaaaabaaaaabbaaaabb$ and
+replacement string $s_2 = c$ yields the string
 
 \[
 ccbcabcaccc