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\section*{Coursework 8 (Scala, Regular Expressions}

This coursework is worth 10\%. It is about regular expressions and

pattern matching. The first part is due on 30 November at 11pm; the

second, more advanced part, is due on 7 December at 11pm. The

second part is not yet included. For the first part you are

asked to implement a regular expression matcher. Make sure the files

you submit can be processed by just calling \texttt{scala

  <<filename.scala>>}.\bigskip

\noindent

\textbf{Important:} Do not use any mutable data structures in your

\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

functions you submit are defined on the ``top-level'' of Scala, not

inside a class or object.

\subsection*{Disclaimer!!!!!!!!}

It should be understood that the work you submit represents

your own effort! You have not copied from anyone else. An

exception is the Scala code I showed during the lectures or

uploaded to KEATS, which you can freely use.\bigskip

\subsection*{Part 1 (6 Marks)}

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

in the file re.scala:

\begin{center}

\begin{tabular}{lcll}

  $r$ & $::=$ & $\ZERO$     & cannot match anything\\

      &   $|$ & $\ONE$      & can only match the empty string\\

      &   $|$ & $c$         & can match a character (in this case $c$)\\

      &   $|$ & $r_1 + r_2$ & can match a string either with $r_1$ or with $r_2$\\

  &   $|$ & $r_1\cdot r_2$ & can match the first part of a string with $r_1$ and\\

          &  & & then the second part with $r_2$\\

      &   $|$ & $r^*$       & can match zero or more times $r$\\

\end{tabular}

\end{center}

\noindent 

Why? Knowing how to match regular expressions and strings fast will

let you solve a lot of problems that vex other humans. Regular

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 over problems such as recently reported at

{\small

\begin{itemize}

\item[$\bullet$] \url{http://stackstatus.net/post/147710624694/outage-postmortem-july-20-2016}

\item[$\bullet$] \url{https://vimeo.com/112065252}

\item[$\bullet$] \url{http://davidvgalbraith.com/how-i-fixed-atom/}  

\end{itemize}}

\subsection*{Tasks (file re.scala)}

\begin{itemize}

\item[(1a)] Implement a function, called \textit{nullable}, by recursion over

  regular expressions. This function tests whether a regular expression can match

  the empty string.

\begin{center}

\begin{tabular}{lcl}

$\textit{nullable}(\ZERO)$ & $\dn$ & $\textit{false}$\\

$\textit{nullable}(\ONE)$  & $\dn$ & $\textit{true}$\\

$\textit{nullable}(c)$     & $\dn$ & $\textit{false}$\\

$\textit{nullable}(r_1 + r_2)$ & $\dn$ & $\textit{nullable}(r_1) \vee \textit{nullable}(r_2)$\\

$\textit{nullable}(r_1 \cdot r_2)$ & $\dn$ & $\textit{nullable}(r_1) \wedge \textit{nullable}(r_2)$\\

$\textit{nullable}(r^*)$ & $\dn$ & $\textit{true}$\\

\end{tabular}

\end{center}\hfill[1 Mark]

\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 according

  to the rules:

\begin{center}

\begin{tabular}{lcl}

$\textit{der}\;c\;(\ZERO)$ & $\dn$ & $\ZERO$\\

$\textit{der}\;c\;(\ONE)$  & $\dn$ & $\ZERO$\\

$\textit{der}\;c\;(d)$     & $\dn$ & $\textit{if}\; c = d\;\textit{then} \;\ONE \; \textit{else} \;\ZERO$\\

$\textit{der}\;c\;(r_1 + r_2)$ & $\dn$ & $(\textit{der}\;c\;r_1) + (\textit{der}\;c\;r_2)$\\

$\textit{der}\;c\;(r_1 \cdot r_2)$ & $\dn$ & $\textit{if}\;\textit{nullable}(r_1)$\\

      & & $\textit{then}\;((\textit{der}\;c\;r_1)\cdot r_2) + (\textit{der}\;c\;r_2)$\\

      & & $\textit{else}\;(\textit{der}\;c\;r_1)\cdot r_2$\\

$\textit{der}\;c\;(r^*)$ & $\dn$ & $(\textit{der}\;c\;r)\cdot (r^*)$\\

\end{tabular}

\end{center}

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 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{4mm}}c@{\hspace{4mm}}ll}

$r \cdot \ZERO$ & $\mapsto$ & $\ZERO$\\ 

$\ZERO \cdot r$ & $\mapsto$ & $\ZERO$\\ 

$r \cdot \ONE$ & $\mapsto$ & $r$\\ 

$\ONE \cdot r$ & $\mapsto$ & $r$\\ 

$r + \ZERO$ & $\mapsto$ & $r$\\ 

$\ZERO + r$ & $\mapsto$ & $r$\\ 

$r + r$ & $\mapsto$ & $r$\\ 

\end{tabular}

  \end{center}

  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 and a regular expression as arguments, and

  builds the derivative w.r.t.~the list as follows:

\begin{center}

\begin{tabular}{lcl}

$\textit{ders}\;(Nil)\;r$ & $\dn$ & $r$\\

  $\textit{ders}\;(c::cs)\;r$  & $\dn$ &

    $\textit{ders}\;cs\;(\textit{simp}(\textit{der}\;c\;r))$\\

\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 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 given the

regular expression $(a\cdot b)\cdot c$ and the string $abc$.

\hfill[1 Mark]

\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 $s_1$ matched by $r$

are replaced by $s_2$. For example given the regular expression

\[(a \cdot a)^* + (b \cdot b)\]

\noindent the string $s_1 = aabbbaaaaaaabaaaaabbaaaabb$ and

replacement string $s_2 = c$ yields the string

\[

ccbcabcaccc

\]

\hfill[2 Mark]

\end{itemize}

\subsection*{Part 2 (4 Marks)}

Coming soon.

\end{document}

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