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

\noindent

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

submission! They are not needed. This excludes 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

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}}

\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

  \[(r_1 + \ZERO) \cdot \ONE + ((\ONE + r_2) + r_3) \cdot (r_4 \cdot \ZERO)\]

\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}

Note that this function is different from \textit{der}, which only

takes a single character.

The second function, 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

$abc$.\\  \mbox{}\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 the string $s_2 = c$ yields the string

\[

ccbcabcaccc

\]

\hfill[2 Marks]

\end{itemize}

\subsection*{Part 2 (4 Marks)}

You need to copy all the code from \texttt{re.scala} into

\texttt{re2.scala} in order to complete Part 2.  Parts (2a) and (2b)

give you another method and datapoints for testing the \textit{der}

and \textit{simp} functions from Part~1.

\begin{itemize}

\item[(2a)] Write a \textbf{polymorphic} function, called

  \textit{iterT}, that is \textbf{tail-recursive}(!) and takes an

  integer $n$, a function $f$ and an $x$ as arguments. This function

  should iterate $f$ $n$-times starting with the argument $x$, like

  \[\underbrace{f(\ldots (f}_{n\text{-times}}(x)))

  \]

  More formally that means \textit{iterT} behaves as follows:

  \begin{center}

    \begin{tabular}{lcl}

      $\textit{iterT}(n, f, x)$ & $\dn$ &

      $\begin{cases}

        \;x & \textit{if}\;n = 0\\

        \;f(\textit{iterT}(n - 1, f, x)) & \textit{otherwise}

        \end{cases}$      

    \end{tabular}

\end{center}

  Make sure you write a \textbf{tail-recursive} version of

  \textit{iterT}.  If you add the annotation \texttt{@tailrec} (see

  \begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-1mm]  

  import scala.annotation.tailrec

  @tailrec

  def iterT[A](n: Int, f: A => A, x: A): A = ...

  \end{lstlisting}

  You can assume that \textit{iterT} will only be called for positive

  integers $0 \le n$. Given the type variable \texttt{A}, the type of

  $f$ is \texttt{A => A} and the type of $x$ is \texttt{A}. This means

  \textit{iterT} can be used, for example, for functions from integers

\item[(2b)] Implement a function, called \textit{size}, by recursion

  over regular expressions. If a regular expression is seen as a tree,

  then \textit{size} should return the number of nodes in such a

  tree. Therefore this function is defined as follows:

\begin{center}

\begin{tabular}{lcl}

$\textit{size}(\ZERO)$ & $\dn$ & $1$\\

$\textit{size}(\ONE)$  & $\dn$ & $1$\\

$\textit{size}(c)$     & $\dn$ & $1$\\

$\textit{size}(r_1 + r_2)$ & $\dn$ & $1 + \textit{size}(r_1) + \textit{size}(r_2)$\\

$\textit{size}(r_1 \cdot r_2)$ & $\dn$ & $1 + \textit{size}(r_1) + \textit{size}(r_2)$\\

$\textit{size}(r^*)$ & $\dn$ & $1 + \textit{size}(r)$\\

\end{tabular}

\end{center}

You can use \textit{size} and \textit{iterT} in order to test how much

the 'evil' regular expression $(a^*)^* \cdot b$ grows when taking

taking the derivative, but simlifying the derivative after each step.

For example, the calls

  \begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-1mm]  

  size(iterT(20, (r: Rexp) => der('a', r), EVIL))

  size(iterT(20, (r: Rexp) => simp(der('a', r)), EVIL))

  \end{lstlisting}

  produce without simplification a regular expression of size of

  just 8.\\ \mbox{}\hfill[1 Mark]

\item[(2c)] Write a \textbf{polymorphic} function, called

  \textit{fixpT}, that takes

  a function $f$ and an $x$ as arguments. The purpose

  of \textit{fixpT} is to calculate a fixpoint of the function $f$

  starting from the argument $x$.

  A fixpoint, say $y$, is when $f(y) = y$ holds. 

  That means \textit{fixpT} behaves as follows:

  \begin{center}

    \begin{tabular}{lcl}

      $\textit{fixpT}(f, x)$ & $\dn$ &

      $\begin{cases}

        \;x & \textit{if}\;f(x) = x\\

        \;\textit{fixpT}(f, f(x)) & \textit{otherwise}

        \end{cases}$      

    \end{tabular}

\end{center}

  Make sure you calculate in the code of $\textit{fixpT}$ the result

  of $f(x)$ only once. Given the type variable \texttt{A} in

  $\textit{fixpT}$, the type of $f$ is \texttt{A => A} and the type of

  $x$ is \texttt{A}. The file \texttt{re2.scala} gives two example

  function where in one the fixpoint is 1 and in the other

  it is the string $a$.\\ \mbox{}\hfill[1 Mark]  

\end{itemize}\bigskip  

\noindent

\textbf{Background} Although easily implementable in Scala, the idea

behind the derivative function might not so easy to be seen. To

understand its purpose better, assume a regular expression $r$ can

match strings of the form $c::cs$ (that means strings which start with

a character $c$ and have some rest, or tail, $cs$). If you now take the

derivative of $r$ with respect to the character $c$, then you obtain a

regular expressions that can match all the strings $cs$.  In other

words, the regular expression $\textit{der}\;c\;r$ can match the same

strings $c::cs$ that can be matched by $r$, except that the $c$ is

chopped off.

Assume now $r$ can match the string $abc$. If you take the derivative

according to $a$ then you obtain a regular expression that can match

$bc$ (it is $abc$ where the $a$ has been chopped off). If you now

build the derivative $\textit{der}\;b\;(\textit{der}\;a\;r))$ you

obtain a regular expression that can match the string $c$ (it is $bc$

where $b$ is chopped off). If you finally build the derivative of this

according $c$, that is

$\textit{der}\;c\;(\textit{der}\;b\;(\textit{der}\;a\;r)))$, you

obtain a regular expression that can match the empty string. You can

test this using the function nullable, which is what your matcher is

doing.

The purpose of the simp function is to keep the regular expression

small. Normally the derivative function makes the regular expression

bigger (see the SEQ case and the example in (2b)) and the algorithm

would be slower and slower over time. The simp function counters this

increase in size and the result is that the algorithm is fast

throughout.  By the way, this algorithm is by Janusz Brzozowski who

came up with the idea of derivatives in 1964 in his PhD thesis.

\begin{center}\small

\url{https://en.wikipedia.org/wiki/Janusz_Brzozowski_(computer_scientist)}

\end{center}

\end{document}

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