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\section*{Handout 1}

This course is about processing of strings. Lets start with what we mean by \emph{string}. Strings
are lists of characters drawn from an \emph{alphabet}. If nothing else is specified, we usually assume 
the alphabet are letters $a$, $b$, \ldots, $z$ and $A$, $B$, \ldots $Z$. Sometimes we explicitly
restrict strings to only contain the letters $a$ and $b$. Then we say the alphabet is the set $\{a, b\}$.

There are many ways how we write string. Since they are lists of characters we might write
them as {\it "hello"} being enclosed by double quotes. This is a short-hand for the list

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

\noindent
The important point is that we can always decompose strings. For example we often consider the
first character of a string, say $h$, and the ``rest''  of a string {\it "ello"}. 
There are also some subtleties with the empty string, sometimes written as {\it ""} or as the empty list
of characters $[\,]$. 

We often need to talk about sets of strings. For example the set of all strings

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

\noindent
Any set of strings, not just the set of all strings, is often called a \emph{language}. The idea behind
this choice 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 string, and so on. In the context of this course, a language might 
not necessarily make sense from a natural language perspective. For example
the set of all strings from 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, or language, that 
contains the empty string $\{\text{""}\}$: the former has no elements, the latter has one 
element.

As seen there are languages which contain infinitely many strings, like the set of all strings.
The ``natural'' languages English, French and so on contain many but only finitely many 
strings (the ones listed in a good dictionary). It might be therefore surprising that the
language consisting of all email addresses is infinite if 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
The reason is that for example before the $@$-sign there can be any string you want if it 
is made up from letters, digits, underscore, dot and hyphen---there are infinitely many
of those. Similarly the string after the $@$-sign can be any string. 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, since 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

\noindent
\emph{Regular expressions} are meant for conveniently describing 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/strings like in a dictionary. 
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 / "" / []\\
        & $\mid$ & $c$                         & 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
There are some subtleties you should be aware of. First, we will use parentheses to disambiguate
regular expressions. For example we will write $(r_1 + r_2)^*$, which is different from $r_1 + (r_2)^*$.
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 a regular expression different from $r_1 + (r_2 + r_3)$,
but in case of $+$ and $\cdot$ we actually do not care 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 $r_1\mid{}r_2$. In case of $\cdot$ we will even often omit it all together. For example
the regular expression for email addresses 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 top-level domain.


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