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\section*{Handout 1}

(they are also sometimes referred to as \emph{words}) are lists of characters drawn from an \emph{alphabet}. 

If nothing else is specified, we usually assume 

the alphabet consists of just the lower-case letters $a$, $b$, \ldots, $z$. Sometimes, however, we explicitly

restrict strings to contain, for example, only the letters $a$ and $b$. In this case we say the alphabet is the set $\{a, b\}$.

There are many ways how we can write down strings. In programming languages, they are usually 

written as {\it "hello"} where the double quotes indicate that we dealing with a string. 

Essentially, strings are lists of characters which can be written for example as follows

\[

[\text{\it h, e, l, l, o}]

\]

\noindent

The important point is that we can always decompose strings. For example, we will often consider the

first character of a string, say $h$, and the ``rest''  of a string say {\it "ello"} when making definitions 

about strings. There are some subtleties with the empty string, sometimes written as {\it ""} but also as 

the empty list of characters $[\,]$. Two strings, for example $s_1$ and $s_2$, can be \emph{concatenated}, 

which we write as $s_1 @ s_2$. Suppose we are given two strings {\it "foo"} and {\it "bar"}, then their concatenation

gives {\it "foobar"}.

We often need to talk about sets of strings. For example the set of all strings over the alphabet $\{a, \ldots\, z\}$

is

\[

\{\text{\it "", "a", "b", "c",\ldots,"z", "aa", "ab", "ac", \ldots, "aaa", \ldots}\}

\]

\noindent

Any set of strings, not just the set-of-all-strings, is often called a \emph{language}. The idea behind

this choice of terminology is that if we enumerate, say, all words/strings from a dictionary, like 

\[

\{\text{\it "the", "of", "milk", "name", "antidisestablishmentarianism", \ldots}\}

\]

\noindent

then we have essentially described the English language, or more precisely all

strings that can be used in a sentence of the English language. French would be a

different set of strings, and so on. In the context of this course, a language might 

not necessarily make sense from a natural language point of view. For example

the set of all strings shown above is a language, as is the empty set (of strings). The

empty set of strings is often written as $\varnothing$ or $\{\,\}$. Note that there is a 

difference between the empty set, or empty language, and the set that 

contains only the empty string $\{\text{""}\}$: the former has no elements, whereas 

the latter has one element.

As seen, there are languages which contain infinitely many strings, like the set of all strings.

The ``natural'' languages like English, French and so on contain many but only finitely many 

strings (namely the ones listed in a good dictionary). It might be therefore be surprising that the

language consisting of all email addresses is infinite provided we assume it is defined by the

regular expression\footnote{See \url{http://goo.gl/5LoVX7}}

\[

([\text{\it{}a-z0-9\_.-}]^+)@([\text{\it a-z0-9.-}]^+).([\text{\it a-z.}]^{\{2,6\}})

\]

\noindent

One reason is that before the $@$-sign there can be any string you want assuming it 

is made up from letters, digits, underscores, dots and hyphens---clearly there are infinitely many

of those. Similarly the string after the $@$-sign can be any string. However, this does not mean 

that every string is an email address. For example

\[

"\text{\it foo}@\text{\it bar}.\text{\it c}"

\]

\noindent

is not, because the top-level-domains must be of length of at least two. (Note that there is

the convention that uppercase letters are treated in email-addresses as if they were

lower-case.)

\bigskip

Before we expand on the topic of regular expressions, let us review some operations on

sets. We will use capital letters $A$, $B$, $\ldots$ to stand for sets of strings. 

The union of two sets is written as usual as $A \cup B$. We also need to define the

operation of \emph{concatenating} two sets of strings. This can be defined as

\[

A @ B \dn \{s_1@ s_2 | s_1 \in A \wedge s_2 \in B \}

\]

\noindent

which essentially means take the first string from the set $A$ and concatenate it with every

string in the set $B$, then take the second string from $A$ do the same and so on. You might

like to think about what this definition means in case $A$ or $B$ is the empty set.

We also need to define

the power of a set of strings, written as $A^n$ with $n$ being a natural number. This is defined inductively as follows

\begin{center}

\begin{tabular}{rcl}

$A^0$ & $\dn$ & $\{[\,]\}$ \\

$A^{n+1}$ & $\dn$ & $A @ A^n$\\

\end{tabular}

\end{center}

\noindent

Finally we need the \emph{star} of a set of strings, written $A^*$. This is defined as the union

of every power of $A^n$ with $n\ge 0$. The mathematical notation for this operation is

\[

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

\]

\noindent

This definition implies that the star of a set $A$ contains always the empty string (that is $A^0$), one 

copy of every string in $A$ (that is $A^1$), two copies in $A$ (that is $A^2$) and so on. In case $A=\{"a"\}$ we therefore 

have 

\[

A^* = \{"", "a", "aa", "aaa", \ldots\}

\]

\noindent

Be aware that these operations sometimes have quite non-intuitive properties, for example

\begin{center}

\begin{tabular}{@{}ccc@{}}

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

$A \cup \varnothing$ & $=$ & $A$\\

$A \cup A$ & $=$ & $A$\\

$A \cup B$ & $=$ & $B \cup A$\\

\end{tabular} &

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

$A @ B$ & $\not =$ & $B @ A$\\

$A  @ \varnothing$ & $=$ & $\varnothing @ A = \varnothing$\\

$A  @ \{""\}$ & $=$ & $\{""\} @ A = A$\\

\end{tabular} &

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

$\varnothing^*$ & $=$ & $\{""\}$\\

$\{""\}^*$ & $=$ & $\{""\}$\\

$A^\star$ & $=$ & $\{""\} \cup A\cdot A^*$\\

\end{tabular} 

\end{tabular}

\end{center}

\bigskip

\noindent

\emph{Regular expressions} are meant to conveniently describe languages...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 all permitted email addresses, as seen above. 

Regular expressions are given by the following grammar:

\begin{center}

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

  $r$ & $::=$ &   $\varnothing$         & null\\

        & $\mid$ & $\epsilon$              & empty string / "" / []\\

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

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

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

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

  \end{tabular}

\end{center}

\noindent

Because we overload our notation, there are some subtleties you should be aware of. First, the letter 

$c$ stands for any character from the

alphabet at hand. Second, we will use parentheses to disambiguate

regular expressions. 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

$r_2$. 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 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 following the convention in the literature, we will in case of $\cdot$ even 

often omit it all together. For example the regular expression for email addresses shown above is meant to be of the form

\[

([\ldots])^+ \cdot @ \cdot ([\ldots])^+ \cdot . \cdot \ldots

\]

\noindent

meaning first comes a name (specified by the regular expression $([\ldots])^+$), then an $@$-sign, then

a domain name (specified by the regular expression $([\ldots])^+$), then a dot and then a top-level domain. Similarly if

we want to specify the regular expression for the string {\it "hello"} we should write

\[

{\it h} \cdot {\it e} \cdot {\it l} \cdot {\it l} \cdot {\it o}

\]

\noindent

but often just write {\it hello}.

Another source of confusion might arise from the fact that we use the term \emph{regular expression} for the regular expressions used in ``theory''

and also the ones used in ``practice''. In this course we refer by default to the regular expressions defined by the grammar above. 

In ``practice'' we often use $r^+$ to stand for one or more times, $\backslash{}d$ to stand for a digit, $r^?$ to stand for an optional regular

expression, or ranges such as $[\text{\it a - z}]$ to stand for any lower case letter from $a$ to $z$. They are however mere convenience 

as they can be seen as shorthand for

\begin{center}

\begin{tabular}{rcl}

$r^+$ & $\mapsto$ & $r\cdot r^*$\\

$r^?$ & $\mapsto$ & $\epsilon + r$\\

$\backslash d$ & $\mapsto$ & $0 + 1 + 2 + \ldots + 9$\\

$[\text{\it a - z}]$ & $\mapsto$ & $a + b + \ldots + z$\\

\end{tabular}

\end{center}

We will see later that the \emph{not}-regular-expression can also be seen as convenience. This regular

expression is supposed to stand for every string \emph{not} matched by a regular expression. We will write

such not-regular-expressions as $\sim{}r$. While being ``convenience'' it is often not so clear what the shorthand for

these kind of not-regular-expressions is. Try to write down the regular expression which can match any

string except the two strings {\it "hello"} and {\it "world"}. It is possible in principle, but often it is easier to just include

$\sim{}r$ in the definition of regular expressions. Whenever we do so, we will state it explicitly.

So far we have only considered informally what the \emph{meaning} of a regular expression is.  

To do so more formally we will associate with every regular expression a set of strings 

that is supposed to be matched by this

regular expression. This can be defined recursively  as follows

\begin{center}

\begin{tabular}{rcl}

$L(\varnothing)$  & $\dn$ & $\{\,\}$\\

$L(\epsilon)$       & $\dn$ & $\{""\}$\\

$L(c)$                  & $\dn$ & $\{"c"\}$\\

$L(r_1+ r_2)$      & $\dn$ & $L(r_1) \cup L(r_2)$\\

$L(r_1 \cdot r_2)$  & $\dn$ & $\{s_1@ s_2 | s_1 \in L(r_1) \wedge s_2 \in L(r_2) \}$\\

$L(r^*)$                   & $\dn$ & $\bigcup_{n \ge 0} L(r)^n$\\

\end{tabular}

\end{center}

\noindent

As a result we can now precisely state what the meaning, for example, of the regular expression 

${\it h} \cdot {\it e} \cdot {\it l} \cdot {\it l} \cdot {\it o}$ is, namely 

$L({\it h} \cdot {\it e} \cdot {\it l} \cdot {\it l} \cdot {\it o}) = \{\text{\it"hello"}\}$...as expected. Similarly if we have the 

choice-regular-expression $a + b$, its meaning is $L(a + b) = \{\text{\it"a"}, \text{\it"b"}\}$, namely the only two strings which can possibly

be matched by this choice. You can now also see why we do not make a difference

between the different regular expressions $(r_1 + r_2) + r_3$ and $r_1 + (r_2 + r_3)$....they 

are not the same regular expression, but have the same meaning. 

The point of the definition of $L$ is that we can use it to precisely specify when a string $s$ is matched by a

regular expression $r$, namely only when $s \in L(r)$. In fact we will write a program {\it match} that takes any string $s$ and

any regular expression $r$ as argument and returns \emph{yes}, if $s \in L(r)$ and \emph{no},

if $s \not\in L(r)$. We leave this for the next lecture.

\begin{figure}[p]

{\lstset{language=Scala}\texttt{\lstinputlisting{../progs/crawler1.scala}}}

\caption{Scala code for a web-crawler that can detect broken links in a web-page. It uses

the regular expression {\tt http\_pattern} in Line~15 for recognising URL-addresses. It finds

all links using the library function {\tt findAllIn} in Line~21.}

\end{figure}

\begin{figure}[p]

{\lstset{language=Scala}\texttt{\lstinputlisting{../progs/crawler2.scala}}}

\caption{A version of the web-crawler which only follows links in ``my'' domain---since these are the

ones I am interested in to fix. It uses the regular expression {\tt my\_urls} in Line~16.

The main change is in Line~26 where there is a test whether URL is in ``my'' domain or not.}

\end{figure}

\begin{figure}[p]

{\lstset{language=Scala}\texttt{\lstinputlisting{../progs/crawler3.scala}}}

\caption{A small email harvester---whenever we download a web-page, we also check whether

it contains any email addresses. For this we use the regular expression {\tt email\_pattern} in

Line~17.  The main change is in Lines 33 and 34 where all email addresses that can be found in a page are printed.}

\end{figure}

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