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\section*{A Crash-Course on Notation}
\subsubsection*{Characters and Strings}
In this module we will often use \defn{characters}. While they
are surely familiar, we will make one subtle distinction. If
we want to refer to concrete characters, like \code{a},
\code{b} and so on, we use a typewriter font. So if we want to
refer to the concrete characters of my email address we shall
write
\begin{center}
\pcode{christian.urban@kcl.ac.uk}
\end{center}
\noindent If we need to explicitly indicate the ``space''
character, we write \VS{}\hspace{1mm}. For example
\begin{center}
\tt{}hello\VS\hspace{0.5mm}world
\end{center}
\noindent But often we do not care
about which characters we use. In such cases we us the italic
font and write $a$, $b$ and so on. So if we need a
representative string, we might write
\begin{equation}\label{abracadabra}
abracadabra
\end{equation}
\noindent We do not really care what the characters stand for,
except we do care about is that for example the character $a$
is not equal to $b$.
An \defn{alphabet} is a finite set of characters. Often the
letter $\Sigma$ is used to refer to an alphabet. For example
the ASCII characters \pcode{a} to \pcode{z} form an alphabet.
The digits $0$ to $9$ are another 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 $\{a, b\}$.
\defn{Strings} are lists of characters. Unfortunately, there
are many ways how we can write down strings. In programming
languages, they are usually written as \dq{$hello$} where the
double quotes indicate that we dealing with a string. But
since, strings are lists of characters we could also write
this string as
\[
[\text{\it h, e, l, l, o}]
\]
\noindent The important point is that we can always decompose
such strings. For example, we will often consider the first
character of a string, say $h$, and the ``rest'' of a string
say \dq{\textit{ello}} when making definitions about strings.
There are some subtleties with the empty string, sometimes
written as \dq{} 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 \dq{\textit{foo}} and \dq{\textit{bar}},
then their concatenation, writen
\dq{\textit{foo}} $@$ \dq{\textit{bar}}, gives
\dq{\textit{foobar}}. Often we will simplify our life and just
drop the double quotes whenever it is clear we are talking
about strings, writing as already in \eqref{abracadabra} just
\textit{foo}, \textit{bar}, \textit{foobar} or \textit{foo $@$
bar}.
Some simple properties of string concatenation hold. For
example the concatenation operation is \emph{associative},
meaning
\[(s_1 @ s_2) @ s_3 = s_1 @ (s_2 @ s_3)\]
\noindent are always equal strings. The empty string behaves
like a unit element, therefore
\[s \,@\, [] = [] \,@\, s = s\]
While for us strings are just lists of characters, programming
languages often differentiate between the two concepts. In
Scala, for example, there is the type of \code{String} and the
type of lists of characters, \code{List[Char]}. They are not
the same and we need to explicitly coerce elements between the
two types, for example
\begin{lstlisting}[numbers=none]
scala> "abc".toList
res01: List[Char] = List(a, b, c)
\end{lstlisting}
\subsubsection*{Sets and Languages}
We will use the familiar operations $\cup$ and $\cap$ for
sets. For the empty set we will either write $\varnothing$ or
$\{\,\}$. The set containing, for example, the natural numbers
$1$, $2$ and $3$ we will write as
\[
\{1, 2, 3\}
\]
\noindent The notation $\in$ means \emph{element of}, so $1
\in \{1, 2, 3\}$ is true and $3 \in \{1, 2, 3\}$ is false.
Sets can potentially have infinitely many elements. For
example the set of all natural numbers $\{0, 1, 2, \ldots\}$
is infinite. This set is often also abbreviated as
$\mathbb{N}$. We can define sets by giving all elements, like
$\{0, 1\}$, but also by \defn{set comprehensions}. For example
the set of all even natural numbers can be defined as
\[
\{n\;|\;n\in\mathbb{N} \wedge n\;\text{is even}\}
\]
\noindent Though silly, but the set $\{0, 1, 2\}$ could also be
defined by the following set comprehension
\[
\{n\;|\; n^2 < 9 \wedge n \in \mathbb{N}\}
\]
\noindent Notice that set comprehensions could be used
to define set union, intersection and difference:
\begin{eqnarray*}
A \cup B & \dn & \{x\;|\; x \in A \vee x \in B\}\\
A \cap B & \dn & \{x\;|\; x \in A \wedge x \in B\}\\
A \backslash B & \dn & \{x\;|\; x \in A \wedge x \not\in B\}
\end{eqnarray*}
\noindent For defining sets, we will also often use the notion
of the ``big union''. An example is as follows:
\begin{equation}\label{bigunion}
\bigcup_{0\le n}\; \{n^2, n^2 + 1\}
\end{equation}
\noindent which is the set of all squares and their immediate
successors, so
\[
\{0, 1, 2, 4, 5, 9, 10, 16, 17, \ldots\}
\]
\noindent A big union is a sequence of unions which are
indexed typically by a natural number. So the big union in
\eqref{bigunion} could equally be written as
\[
\{0, 1\} \cup \{1, 2\} \cup \{4, 5\} \cup \{9, 10\} \cup
\ldots
\]
\noindent but using the big union notation is more concise.
An important notion in this module are \defn{Languages}, which
are sets of strings. The main goal for us will be how to
(formally) specify languages and to find out whether a string
is in a language or not. Note that the language containing the
empty string $\{\dq{}\}$ is not equal to the empty language
(or empty set): The former contains one element, namely \dq{}
(also written $[\,]$), but the latter does not contain any.
For languages we define the operation of \defn{language
concatenation}, written $A @ B$:
\begin{equation}\label{langconc}
A @ B \dn \{s_1 @ s_2\;|\; s_1\in A \wedge s_2\in B\}
\end{equation}
\noindent Be careful to understand the difference: the $@$
in $s_1 @ s_2$ is string concatenation, while $A @ B$ refers
to the concatenation of two languages (or sets of strings).
As an example suppose $A=\{ab, ac\}$ and $B=\{zzz, qq, r\}$,
then $A \,@\, B$ is
\[
\{abzzz, abqq, abr, aczzz, acqq, acr\}
\]
\noindent Recall the properties for string concatenation. For
language concatenation we have the following properties
\begin{center}
\begin{tabular}{ll}
associativity: & $(A @ B) @ C = A @ (B @ C)$\\
unit element: & $A \,@\, \{[]\} = \{[]\} \,@\, A = A$\\
zero element: & $A \,@\, \varnothing = \varnothing \,@\, A =
\varnothing$
\end{tabular}
\end{center}
\noindent Note the difference: the empty set behaves like $0$
for multiplication and the set $\{[]\}$ like $1$ for
multiplication.
Following the language concatenation, we can define a
\defn{language power} operation as follows:
\begin{eqnarray*}
A^0 & \dn & \{[]\}\\
A^{n+1} & \dn & A \,@\, A^n
\end{eqnarray*}
\noindent This definition is by induction on natural numbers.
Note carefully that the zero-case is not defined as the empty
set, but the set containing the empty string. So no matter
what the set $A$ is, $A^0$ will always be $\{[]\}$. (There is
another hint about a connection between the $@$-operation and
multiplication: How is $x^n$ defined and what is $x^0$?)
Next we can define the \defn{star operation} for languages:
$A^*$ is the union of all powers of $A$, or short
\[
A^* \dn \bigcup_{0\le n}\; A^n
\]
\noindent
Unfolding this definition
\[
A^0 \cup A^1 \cup A^2 \cup A^3 \cup \ldots
\]
\noindent
which is equal to
\[
\{[]\} \,\cup\, A \,\cup\, A @ A \,\cup\, A @ A @ A \,\cup\, \ldots
\]
\noindent we can see that the empty string is always in $A^*$,
no matter what $A$ is. This is because $[] \in A^0$. To make
sure you understand these definition, I leave you to answer
what $\{[]\}^*$ and $\varnothing^*$ are.
Recall that an alphabet is often referred to by the letter
$\Sigma$. We can now write for the set of all strings over
this alphabet $\Sigma^*$. In doing so we also include the
empty string as a possible string over $\Sigma$. So if
$\Sigma = \{a, b\}$ then $\Sigma^*$ is
\[
\{[], a, b, ab, ba, aaa, aab, aba, abb, baa, bab, \ldots\}
\]
\noindent or in other words all strings containing $a$s and
$b$s only.
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