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\section*{A Crash-Course on Notation}

There are innumerable books available about compilers,

automata theory and formal languages. Unfortunately, they

often use their own notational conventions and their own

symbols. This handout is meant to clarify some of the notation

I will use. I appologise in advance that sometimes I will be a

bit fuzzy\ldots the problem is that often we want to have

convenience in our mathematical definitions (to make them

readable and understandable), but sometimes we need precision

for actual programs.

\subsubsection*{Characters and Strings}

The most important concept in this module are strings. Strings

are composed of \defn{characters}. While characters are surely

a familiar concept, we will make one subtle distinction in

this module. If we want to refer to concrete characters, like

\code{a}, \code{b}, \code{c} and so on, we use a typewriter font.

Accordingly 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 also 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 which particular characters

we use. In such cases we use the italic font and write $a$,

$b$, $c$ and so on for characters. Therefore if we need a

representative string, we might write

\[

abracadabra

\]

\noindent In this string, we do not really care what the

characters stand for, except we do care about the fact that

for example the character $a$ is not equal to $b$ and so on.

Why do I make this distinction? Because we often need to

define functions using variables ranging over characters. We

need to somehow say this is a variable, say $c$, ranging over

characters, while this is the atual character \pcode{c}.

An \defn{alphabet} is a (non-empty) 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. The

Greek letters $\alpha$ to $\omega$ also form an 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 want to restrict strings to

contain only the letters $a$ and $b$, for example. In this

case we will state that 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 are dealing with a string. In

programming languages, such as Scala, strings have a special

type---namely \pcode{String} which is different from the type

for lists of chatacters. This is because strings can be

efficiently represented in memory, unlike general lists. Since

\code{String} and the type of lists of characters,

\code{List[Char]} are not the same, 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}

\noindent Since in our (mathematical) definitions we regard

strings as lists of characters, we will also write

\dq{$hello$} as

\[

[\text{\it h, e, l, l, o}] \qquad\text{or simply}\qquad \textit{hello}

\]

\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 also some subtleties with the empty string,

sometimes written as \dq{} but also as the empty list of

characters $[\,]$.\footnote{In the literature you can also

often find that $\varepsilon$ or $\lambda$ is used to

represent the empty string.} 

Two strings, say $s_1$ and $s_2$, can be \defn{concatenated},

which we write as $s_1 @ s_2$. If we regard $s_1$ and $s_2$ as

lists of characters, then $@$ is the list-append function.

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}}. But as said above, we will often

simplify our life and just drop the double quotes whenever it

is clear we are talking about strings, So we will often just

write \textit{foo}, \textit{bar}, \textit{foobar} or

\textit{foo $@$ bar}.

Occasionally we will use the notation $a^n$ for strings, which 

stands for the string of $n$ repeated $a$s. So $a^{n}b^{n}$ is

a string that has as many $a$s by as many $b$s.

A simple property of string concatenation is 

\emph{associativity}, meaning

\[(s_1 @ s_2) @ s_3 = s_1 @ (s_2 @ s_3)\]  

\noindent are always equal strings. The empty string behaves

like a \emph{unit element}, therefore

\[s \,@\, [] = [] \,@\, s = s\]

\subsubsection*{Sets and Languages}

We will use the familiar operations $\cup$, $\cap$, $\subset$

and $\subseteq$ for sets. For the empty set we will either

write $\varnothing$ or $\{\,\}$. The set containing the

natural numbers $1$, $2$ and $3$, for example, we will write

with curly braces as

\[

\{1, 2, 3\}

\]

\noindent The notation $\in$ means \emph{element of}, so $1

\in \{1, 2, 3\}$ is true and $4 \in \{1, 2, 3\}$ 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, for

example $\{0, 1\}$ for the set containing just $0$ and $1$,

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 In general set comprehensions are of the form

$\{a\;|\;P\}$ which stands for the set of all elements $a$

(from some set) for which some property $P$ holds.

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.

While this stuff about sete might all look trivial or even

needlessly pedantic, \emph{Nature} is never simple. If you

want to be amazed how complicated sets can get, watch out for

the last lecture just before Christmas where I want you to

convince you of the fact that some sets are more infinite than

others. Actually that will be a fact that is very relevant to

the material of this module.

Another important notion in this module are \defn{languages}, which

are sets of strings. One of the main goals for us will be how to

(formally) specify languages and to find out whether a string

is in a language or not.\footnote{You might wish to ponder

whether this is in general a hard or easy problem, where

hardness is meant in terms of Turing decidable, for example.}

Note that the language containing the empty string $\{\dq{}\}$

is not equal to $\varnothing$, the empty language (or empty

set): The former contains one element, namely \dq{} (also

written $[\,]$), but the latter does not contain any

element.

For languages we define the operation of \defn{language

concatenation}, written like in the string case as $A @ B$:

\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 the language

\[

\{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 in the last two lines: the empty

set behaves like $0$ for multiplication and the set $\{[]\}$

like $1$ for multiplication ($n * 1 = n$ and $n * 0 = 0$).

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 recursion 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 recursively and what is

$x^0$?)

Next we can define the \defn{star operation} for languages:

$A\star$ is the union of all powers of $A$, or short

\begin{equation}\label{star}

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

\end{equation}

\noindent This star operation is often also called

\emph{Kleene-star}. Unfolding the definition in \eqref{star}

gives

\[

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 $A\star$,

no matter what $A$ is. This is because $[] \in A^0$. To make

sure you understand these definitions, I leave you to answer

what $\{[]\}\star$ and $\varnothing\star$ are?

Recall that an alphabet is often referred to by the letter

$\Sigma$. We can now write for the set of \emph{all} strings

over this alphabet $\Sigma\star$. In doing so we also include the

empty string as a possible string over $\Sigma$. So if $\Sigma

= \{a, b\}$, then $\Sigma\star$ is

\[

\{[], a, b, aa, ab, ba, bb, aaa, aab, aba, abb, baa, bab, \ldots\}

\]

\noindent or in other words all strings containing $a$s and

$b$s only, plus the empty string.

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