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\begin{document}

\fnote{\copyright{} Christian Urban, King's College London, 2014, 2015, 2016, 2017, 2018, 2019}

\section*{Handout 1}

pattern---be it an email address, for example. In this way we can

exclude user input that would otherwise have nasty effects on our

program (crashing it or making it go into an infinite loop, if not

filtering out spam usually involves scanning for some signature

Knuth-Morris-Pratt algorithm for strings is not good enough for such

string \emph{patterns}.\smallskip

\defn{Regular expressions} help with conveniently specifying

such patterns. The idea behind regular expressions is that

they are a simple method for describing languages (or sets of

strings)\ldots at least languages we are interested in in

computer science. For example there is no convenient regular

expression for describing the English language short of

enumerating all English words. But they seem useful for

describing for example simple email addresses.\footnote{See

``8 Regular Expressions You Should Know''

\url{http://goo.gl/5LoVX7}} Consider the following regular

expression

\begin{equation}\label{email}

\texttt{[a-z0-9\_.-]+} \;\;\texttt{@}\;\; \texttt{[a-z0-9.-]+} \;\;\texttt{.}\;\; \texttt{[a-z.]\{2,6\}}

\end{equation}

\begin{lstlisting}[language={},numbers=none,keywordstyle=\color{black}]

niceandsimple@example.org

very.common@example.co.uk

a.little.lengthy.but.fine@dept.example.ac.uk

other.email-with-dash@example.edu

\end{lstlisting}

\noindent

But for example the following two do not

\begin{lstlisting}[language={},numbers=none,keywordstyle=\color{black}]

user@localserver

disposable.style.email.with+symbol@example.com

\end{lstlisting}

\noindent according to the regular expression we specified in line

\eqref{email} above. Whether this is intended or not is a different

question (the second email above is actually an acceptable email

address according to the RFC 5322 standard for email addresses).

As mentioned above, identifiers, or variables, in program code

are often required to satisfy the constraints that they start

with a letter and then can be followed by zero or more letters

or numbers and also can include underscores, but not as the

first character. Such identifiers can be recognised with the

regular expression

\begin{center}

\pcode{[a-zA-Z] [a-zA-Z0-9_]*}

\end{center}

\noindent Possible identifiers that match this regular expression 

are \pcode{x}, \pcode{foo}, \pcode{foo_bar_1}, \pcode{A_very_42_long_object_name},

but not \pcode{_i} and also not \pcode{4you}.

Many programming languages offer libraries that can be used to

validate such strings against regular expressions. Also there

are some common, and I am sure very familiar, ways of how to

construct regular expressions. For example in Scala we have

a library implementing the following regular expressions: 

\begin{center}

\begin{tabular}{lp{9cm}}

\pcode{re*} & matches 0 or more occurrences of preceding 

expression\\

\pcode{re+} & matches 1 or more occurrences of preceding

expression\\

\pcode{re?} &	 matches 0 or 1 occurrence of preceding 

expression\\

\pcode{re\{n\}}	& matches exactly \pcode{n} number of 

occurrences of preceding  expression\\

\pcode{re\{n,m\}} & matches at least \pcode{n} and at most {\tt m}

occurrences of the preceding expression\\

\pcode{[...]} & matches any single character inside the 

brackets\\

\pcode{[^...]} & matches any single character not inside the 

brackets\\

\pcode{...-...} & character ranges\\

\pcode{\\d} & matches digits; equivalent to \pcode{[0-9]}\\

\pcode{.} & matches every character except newline\\

\pcode{(re)}	& groups regular expressions and remembers 

matched text

\end{tabular}

\end{center}

\noindent With this table you can figure out the purpose of

the regular expressions in the web-crawlers shown Figures

\ref{crawler1}, \ref{crawler2} and

\ref{crawler3}. Note, however, the regular expression for

http-addresses in web-pages in Figure~\ref{crawler1}, Line 15,

is intended to be

\[

\pcode{"https?://[^"]*"}

\]

\noindent It specifies that web-addresses need to start with a

double quote, then comes \texttt{http} followed by an optional

\texttt{s} and so on until the closing double quote comes.

Usually we would have to escape the double quotes in order to

make sure we interpret the double quote as character, not as

double quote for a string. But Scala's trick with triple

quotes allows us to omit this kind of escaping. As a result we

can just write:

\[

\pcode{""""https?://[^"]*"""".r}

\]

\subsection*{Why Study Regular Expressions?}

Regular expressions were introduced by Kleene in the 1950ies and they

have been object of intense study since then. They are nowadays pretty

much ubiquitous in computer science. There are many libraries

implementing regular expressions. I am sure you have come across them

\begin{center}

\begin{tikzpicture}

\begin{axis}[

    title={Graph: $\texttt{(a*)*\,b}$ and strings 

           $\underbrace{\texttt{a}\ldots \texttt{a}}_{n}$},

    xlabel={$n$},

    x label style={at={(1.05,0.0)}},

    ylabel={time in secs},

    enlargelimits=false,

    xtick={0,5,...,30},

    xmax=33,

    ymax=35,

    ytick={0,5,...,30},

    scaled ticks=false,

    axis lines=left,

    width=5.5cm,

    height=4.5cm, 

    legend entries={Python, Java 8, JavaScript},  

    legend pos=north west,

    legend cell align=left]

\addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data};

\addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data};

\addplot[red,mark=*, mark options={fill=white}] table {re-js.data};

\end{axis}

\end{tikzpicture}

&

\begin{tikzpicture}

\begin{axis}[

    title={Graph: $\texttt{a?\{n\}\,a{\{n\}}}$ and strings 

           $\underbrace{\texttt{a}\ldots \texttt{a}}_{n}$},

    xlabel={$n$},

    x label style={at={(1.05,0.0)}},

    ylabel={time in secs},

    enlargelimits=false,

    xtick={0,5,...,30},

    xmax=33,

    ymax=35,

    ytick={0,5,...,30},

    scaled ticks=false,

    axis lines=left,

    width=5.5cm,

    height=4.5cm, 

    legend entries={Python,Ruby},  

    legend pos=north west,

    legend cell align=left]

\addplot[blue,mark=*, mark options={fill=white}] table {re-python.data};

\addplot[brown,mark=triangle*, mark options={fill=white}] table {re-ruby.data};  

\end{axis}

\end{tikzpicture}

\end{tabular}

\end{center}

\noindent This first graph shows that Python, JavaScript and Java 8 need

approximately 30 seconds to find out that the regular expression

Admittedly, these regular expressions are carefully chosen to exhibit

this exponential behaviour, but similar ones occur more often than one

wants in ``real life''. For example, on 20 July 2016 a similar regular

expression brought the webpage \href{http://stackexchange.com}{Stack

\begin{center}

\url{http://stackstatus.net/post/147710624694/outage-postmortem-july-20-2016}

\end{center}

\noindent I can also highly recommend a cool webtalk from an engineer

from Stack Exchange on the same subject:

\begin{center}

\url{https://vimeo.com/112065252}

\end{center}

\noindent

A similar problem also occurred in the Atom editor:

\begin{center}

\url{http://davidvgalbraith.com/how-i-fixed-atom/}

\end{center}

\noindent

\begin{center}

\url{https://www.tutorialdocs.com/article/regex-trap.html}

\end{center}  

Such troublesome regular expressions are sometimes called \emph{evil

  regular expressions} because they have the potential to make regular

  Denial of Service Attacks (ReDoS)}.

It will be instructive to look behind the ``scenes'' to find

out why Python and Ruby (and others) behave so badly when

matching strings with evil regular expressions. But we will also look

at a relatively simple algorithm that solves this problem much

better than Python and Ruby do\ldots actually it will be two

versions of the algorithm: the first one will be able to

process strings of approximately 1,100 \texttt{a}s in 23 

seconds, while the second version will even be able to process

up to 11,000(!) in 5 seconds, see the graph below:

\begin{center}

\begin{tikzpicture}

  \begin{axis}[

    title={Graph: $\texttt{a?\{n\}\,a{\{n\}}}$ and strings 

           $\underbrace{\texttt{a}\ldots \texttt{a}}_{n}$},

    xlabel={$n$},

    x label style={at={(1.05,0.0)}},

    ylabel={time in secs},

    enlargelimits=false,

    xtick={0,3000,...,12000},

    xmax=13000,

    ymax=32,

    ytick={0,5,...,30},

    scaled ticks=false,

    axis lines=left,

    width=7cm,

    height=4.5cm,

    legend entries={Our Algorithm V1, Our Algorithm V2},

    legend pos=outer north east]

\addplot[green,mark=square*,mark options={fill=white}] table {re2.data};

\addplot[black,mark=square*,mark options={fill=white}] table {re3.data};

\end{axis}

\end{tikzpicture}

\end{center}

\noindent And in the case of the regular expression $\texttt{(a*)*\,b}$

and strings of \texttt{a}s we will beat Java 8 by factor of

approximately 1,000,000 (note the scale on the $x$-axis). 

\begin{center}

\begin{tikzpicture}

  \begin{axis}[

    title={Graph: $\texttt{(a*)*\,b}$ and strings 

           $\underbrace{\texttt{a}\ldots \texttt{a}}_{n}$},

    xlabel={$n$},

    x label style={at={(1.05,0.0)}},

    ylabel={time in secs},

    enlargelimits=false,

    ymax=32,

    ytick={0,5,...,30},

    axis lines=left,

    width=7cm,

    height=4.5cm,

    legend entries={Our Algorithm V2},

    legend pos=outer north east]

\addplot[black,mark=square*,mark options={fill=white}] table {re3a.data};

\end{axis}

\end{tikzpicture}

\end{center}

\subsection*{Basic Regular Expressions}

The regular expressions shown earlier for Scala, we

will call \emph{extended regular expressions}. The ones we

will mainly study in this module are \emph{basic regular

expressions}, which by convention we will just call

\emph{regular expressions}, if it is clear what we mean. The

attraction of (basic) regular expressions is that many

features of the extended ones are just syntactic sugar.

(Basic) regular expressions are defined by the following

grammar:

\begin{center}

\begin{tabular}{r@{\hspace{1mm}}r@{\hspace{1mm}}l@{\hspace{13mm}}l}

  $r$ & $::=$ &    $\ZERO$          & null language\\

        & $\mid$ & $\ONE$           & empty string / \texttt{""} / []\\

        & $\mid$ & $c$                  & single character\\

        & $\mid$ & $r_1 + r_2$          & alternative / choice\\

        & $\mid$ & $r_1 \cdot r_2$      & sequence\\

        & $\mid$ & $r^*$                & star (zero or more)\\

  \end{tabular}

\end{center}

\noindent Because we overload our notation, there are some

subtleties you should be aware of. When regular expressions

are referred to, then $\ZERO$ (in bold font) does not stand for

the number zero: rather it is a particular pattern that does

not match any string. Similarly, in the context of regular

expressions, $\ONE$ does not stand for the number one but for

a regular expression that matches the empty string. The letter

$c$ stands for any character from the alphabet at hand. Again

in the context of regular expressions, it is a particular

pattern that can match the specified character. You should

also be careful with our overloading of the star: assuming you

have read the handout about our basic mathematical notation,

you will see that in the context of languages (sets of

strings) the star stands for an operation on languages. Here

$r^*$ stands for a regular expression, which is different from

the operation on sets is defined as

\[

A\star\dn \bigcup_{0\le n} A^n

\]

We will use parentheses to disambiguate regular expressions.

Parentheses are not really part of a regular expression, and

indeed we do not need them in our code because there the tree

structure of regular expressions is always clear. But for

writing them down in a more mathematical fashion, parentheses

will be helpful. For example we will write $(r_1 + r_2)^*$,

which is different from, say $r_1 + (r_2)^*$. The former means

roughly zero or more times $r_1$ or $r_2$, while the latter

means $r_1$, or zero or more times $r_2$. This will turn out to

be two different patterns, which match in general different

strings. We should also write $(r_1 + r_2) + r_3$, which is

different from the regular expression $r_1 + (r_2 + r_3)$, but

in case of $+$ and $\cdot$ we actually do not care about the

order and just write $r_1 + r_2 + r_3$, or $r_1 \cdot r_2

\cdot r_3$, respectively. The reasons for this carelessness will become

clear shortly. 

In the literature you will often find that the choice $r_1 +

r_2$ is written as $r_1\mid{}r_2$ or $r_1\mid\mid{}r_2$. Also,

often our $\ZERO$ and $\ONE$ are written $\varnothing$ and

$\epsilon$, respectively. Following the convention in the

literature, we will often omit the $\cdot$. This

is to make some concrete regular expressions more readable.

For example the regular expression for email addresses shown

in \eqref{email} would fully expanded look like

\[

\texttt{[...]+} \;\cdot\;  \texttt{@} \;\cdot\; 

\texttt{[...]+} \;\cdot\; \texttt{.} \;\cdot\; 

\texttt{[...]\{2,6\}}

\]

\noindent

which is much less readable than the regular expression in

\eqref{email}. Similarly for the regular expression that matches the

string $hello$ we should write

\[

h \cdot e \cdot l \cdot l \cdot o

\]

\noindent

but often just write {\it hello}.

If you prefer to think in terms of the implementation

of regular expressions in Scala, the constructors and

classes relate as follows\footnote{More about Scala is 

in the handout about \emph{A Crash-Course on Scala}.}

\begin{center}

\begin{tabular}{rcl}

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

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

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

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