% !TEX program = xelatex

\documentclass{article}

\usepackage{../style}

\usepackage{../langs}

\usepackage{disclaimer}

\usepackage{tikz}

\usepackage{pgf}

\usepackage{pgfplots}

\usepackage{stackengine}

%% \usepackage{accents}

\newcommand\barbelow[1]{\stackunder[1.2pt]{#1}{\raisebox{-4mm}{\boldmath$\uparrow$}}}

\begin{filecontents}{re-python2.data}

1 0.033

5 0.036

10 0.034

15 0.036

18 0.059

19 0.084 

20 0.141

21 0.248

22 0.485

23 0.878

24 1.71

25 3.40

26 7.08

27 14.12

28 26.69

\end{filecontents}

\begin{filecontents}{re-java.data}

5  0.00298

10  0.00418

15  0.00996

16  0.01710

17  0.03492

18  0.03303

19  0.05084

20  0.10177

21  0.19960

22  0.41159

23  0.82234

24  1.70251

25  3.36112

26  6.63998

27  13.35120

28  29.81185

\end{filecontents}

\begin{filecontents}{re-js.data}

5   0.061

10  0.061

15  0.061

20  0.070

23  0.131

25  0.308

26  0.564

28  1.994

30  7.648

31  15.881 

32  32.190

\end{filecontents}

\begin{filecontents}{re-java9.data}

1000  0.01410

2000  0.04882

3000  0.10609

4000  0.17456

5000  0.27530

6000  0.41116

7000  0.53741

8000  0.70261

9000  0.93981

10000 0.97419

11000 1.28697

12000 1.51387

14000 2.07079

16000 2.69846

20000 4.41823

24000 6.46077

26000 7.64373

30000 9.99446

34000 12.966885

38000 16.281621

42000 19.180228

46000 21.984721

50000 26.950203

60000 43.0327746

\end{filecontents}

\begin{filecontents}{re-swift.data}

5   0.001

10  0.001

15  0.009

20  0.178

23  1.399

24  2.893

25  5.671

26  11.357

27  22.430

\end{filecontents}

\begin{filecontents}{re-dart.data}

20 0.042

21 0.084

22 0.190

23 0.340

24 0.678

25 1.369

26 2.700

27 5.462

28 10.908

29 21.725

30 43.492

\end{filecontents}

\begin{document}

% BF IDE

% https://www.microsoft.com/en-us/p/brainf-ck/9nblgggzhvq5

%\mbox{}\hfill\textit{``[Google’s MapReduce] abstraction is inspired by the}\\

%\mbox{}\hfill\textit{map and reduce primitives present in Lisp and many}\\

%\mbox{}\hfill\textit{other functional language.''}\smallskip\\

%\mbox{}\hfill\textit{ --- Dean and Ghemawat, who designed this concept at Google}

%\bigskip\medskip

\noindent

This part is about a regular expression matcher described by

Brzozowski in 1964. This part is due on \cwEIGHTa{} at 5pm.  The

background is that ``out-of-the-box'' regular expression matching in

mainstream languages like Java, JavaScript and Python can sometimes be

excruciatingly slow.  You are supposed to implement a regular

expression matcher that is much, much faster. \bigskip

\IMPORTANTNONE{}

\noindent

Also note that the running time of each part will be restricted to a

maximum of 30 seconds on my laptop.  

\DISCLAIMER{}

\subsection*{Reference Implementation}

This Scala assignment comes with a reference implementation in form of

a \texttt{jar}-file. This allows you to run any test cases on your own

computer. For example you can call Scala on the command line with the

option \texttt{-cp re.jar} and then query any function from the

\texttt{re.scala} template file. As usual you have to prefix the calls

are time sensitive, you can check the reference implementation as

follows: if you want to know, for example, how long it takes to match

strings of $a$'s using the regular expression $(a^*)^*\cdot b$ you can

query as follows:

\begin{lstlisting}[xleftmargin=1mm,numbers=none,basicstyle=\ttfamily\small]

$ scala -cp re.jar

scala> for (i <- 0 to 5000000 by 500000) {

  | println(f"$i: ${time_needed(2, matcher(EVIL, "a" * i))}%.5f secs.")

  | }

0: 0.00002 secs.

500000: 0.10608 secs.

1000000: 0.22286 secs.

1500000: 0.35982 secs.

2000000: 0.45828 secs.

2500000: 0.59558 secs.

3000000: 0.73191 secs.

3500000: 0.83499 secs.

4000000: 0.99149 secs.

4500000: 1.15395 secs.

5000000: 1.29659 secs.

\end{lstlisting}%$

\subsection*{Preliminaries}

The task is to implement a regular expression matcher that is based on

derivatives of regular expressions. Most of the functions are defined by

recursion over regular expressions and can be elegantly implemented

using Scala's pattern-matching. The implementation should deal with the

following regular expressions, which have been predefined in the file

\texttt{re.scala}:

\begin{center}

\begin{tabular}{lcll}

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

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

      &   $|$ & $c$         & can match a single 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 a string with zero or more copies of $r$\\

\end{tabular}

\end{center}

\noindent 

Why? Regular expressions are

one of the simplest ways to match patterns in text, and

are endlessly useful for searching, editing and analysing data in all

sorts of places (for example analysing network traffic in order to

detect security breaches). However, you need to be fast, otherwise you

will stumble over problems such as recently reported at

{\small

\begin{itemize}

\item[$\bullet$] \url{https://blog.cloudflare.com/details-of-the-cloudflare-outage-on-july-2-2019}  

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

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

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

\end{itemize}}

% Knowing how to match regular expressions and strings will let you

% solve a lot of problems that vex other humans.

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

The file \texttt{re.scala} has already a definition for regular

expressions and also defines some handy shorthand notation for

regular expressions. The notation in this document matches up

with the code in the file as follows:

\begin{center}

  \begin{tabular}{rcl@{\hspace{10mm}}l}

    & & code: & shorthand:\smallskip \\ 

  $\ZERO$ & $\mapsto$ & \texttt{ZERO}\\

  $\ONE$  & $\mapsto$ & \texttt{ONE}\\

  $c$     & $\mapsto$ & \texttt{CHAR(c)}\\

  $r_1 + r_2$ & $\mapsto$ & \texttt{ALT(r1, r2)} & \texttt{r1 | r2}\\

  $r_1 \cdot r_2$ & $\mapsto$ & \texttt{SEQ(r1, r2)} & \texttt{r1 $\sim$ r2}\\

  $r^*$ & $\mapsto$ &  \texttt{STAR(r)} & \texttt{r.\%}

\end{tabular}    

\end{center}  

\begin{itemize}

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

  recursion over regular expressions. This function tests whether a

  regular expression can match the empty string. This means given a

  regular expression it either returns true or false. The function

  \textit{nullable}

  is defined as follows:

\begin{center}

\begin{tabular}{lcl}

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

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

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

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

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

\end{tabular}

\item[(2)] 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 of a regular expression according

  to the rules:

\begin{center}

\begin{tabular}{lcl}

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

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

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

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

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

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

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

\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}

    $\textit{der}\;a\;r$ & $=$ & $(\ONE \cdot b)\cdot c$ & \quad($= r'$)\\

    $\textit{der}\;b\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$\\

    $\textit{der}\;c\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$

  \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}

    $\textit{der}\;a\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$ \\

    $\textit{der}\;b\;r'$ & $=$ & $((\ZERO \cdot b) + \ONE)\cdot c$ & \quad($= r''$)\\

    $\textit{der}\;c\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$

  \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}

    $\textit{der}\;a\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$ \\

    $\textit{der}\;b\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$\\

    $\textit{der}\;c\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ONE$ &

    (is $\textit{nullable}$)                      

  \end{tabular}

\end{center}

Note, the last derivative can match the empty string, that is it is \textit{nullable}.\\

\mbox{}\hfill\mbox{[1 Mark]}

  traverses a regular expression, and on the way up simplifies every

  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}

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

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

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

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

\end{tabular}

  For example the regular expression

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

  simplifies to just $r_1$. \textbf{Hint:} Regular expressions can be

  seen as trees and there are several methods for traversing

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

$abc$, but false if you give it the string $ab$. \hfill[1 Mark]

  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}

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

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

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

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

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

\end{tabular}

\end{center}

You can use \textit{size} in order to test how much the ``evil'' regular

expression $(a^*)^* \cdot b$ grows when taking successive derivatives

according the letter $a$ without simplification and then compare it to

taking the derivative, but simplify the result.  The sizes

  task.  The purpose here is that you will be marked for some ``power''

  test cases. For example can your matcher decide within 30 seconds

  whether the regular expression $(a^*)^*\cdot b$ matches strings of the

  form $aaa\ldots{}aaaa$, for say 1 Million $a$'s. And does simplification

  simplify the regular expression

  \[

  \texttt{SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE)}

  \]  

  \noindent correctly to just \texttt{ONE}, where \texttt{SEQ} is nested

  50 or more times?\\

\end{itemize}

\subsection*{Background}

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

whether this is indeed the case using the function nullable, which is

what your matcher is doing.

The purpose of the $\textit{simp}$ function is to keep the regular

expressions small. Normally the derivative function makes the regular

expression bigger (see the SEQ case and the example in (2)) and the

algorithm would be slower and slower over time. The $\textit{simp}$

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}