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\fnote{\copyright{} Christian Urban, King's College London, 2014, 2015, 2016, 2017, 2018, 2020}

\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 apologise 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 other times we need pedantic

precision for actual programs.

\subsubsection*{Characters and Strings}

The most basic 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 will 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'' and give it a name. 

In such cases we use the italic font.

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{\texttt{hello}} where the

double quotes indicate that we are dealing with a string. In

typed programming languages, such as Scala, strings have a special

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

for lists of characters. This is because strings can be

efficiently represented in memory, unlike lists. Since

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

(namely \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

However, we do not want to do this kind of explicit coercion in our

pencil-and-paper, everyday arguments.  So in our (mathematical)

definitions we regard strings as lists of characters and we will also

write \dq{$hello$} as list

\[

[\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. But we are not going to use this notation.} 

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, written

\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 just

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

\textit{foo $@$ bar} and so on.

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 some number of $a$s followed by the same number of $b$s.

Confusingly, in Scala the notation is ``times'' for this opration.

So you can write 

\begin{lstlisting}[numbers=none]

scala> "a" * 13

val res02: String = aaaaaaaaaaaaa

\end{lstlisting}

\noindent

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.  Note that the

\emph{list} $[1, 2, 3]$ is something different from the \emph{set}

$\{1, 2, 3\}$: in the former we care about the order and potentially

several occurrences of a number; while with the latter we do not.

Also sets can potentially have infinitely many elements, whereas lists

cannot. For example

the set of all natural numbers $\{0, 1, 2, \ldots\}$ is infinite. This

set is often also abbreviated as $\mathbb{N}$. Lists can be very large, but they cannot contain infinitely many elements.

We can define sets by giving all their elements, for example $\{0, 1\}$ for

the set containing just $0$ and $1$. But often we need to use \defn{set

  comprehensions} to define sets. For example the set of all \emph{even}

natural numbers can be defined as

\[

\{n\;|\;n\in\mathbb{N} \wedge n\;\text{is even}\}

\]

\noindent Set comprehensions consist of a ``base set'' (in this case

all the natural numbers) and a predicate (here eveness).

Though silly, but the set $\{0, 1, 2\}$ could also be

defined by the following set comprehension

\[

\{n\;|\;  n \in \mathbb{N} \wedge n^2 < 9\}

\]

\noindent Can you see why this defines the set $\{0, 1, 2\}$?  Notice

that set comprehensions are quite powerful constructions. For example they

could be used to define set union,

set intersection and set 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. If programming

is more your-kind-of-thing, you might recognise the similarities

with for-comprehensions, for example for the silly set above you

could write

\begin{lstlisting}[numbers=none]

scala> for (n <- (0 to 10).toSet; if n * n < 9) yield n

val res03: Set[Int] = Set(0, 1, 2)

\end{lstlisting}

\noindent

This is pretty much the same as $\{n\;|\;  n \in \mathbb{N} \wedge n^2 < 9\}$

just in Scala syntax.

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.

As an aside: While this stuff about sets 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 to convince you of the fact

that some sets are more infinite than other sets. Yes, you read

correctly, there can be sets that are ``more infinite'' than

others. If you think this is obvious: say you have the infinite set

$\mathbb{N}\backslash\{0\} = \{1, 2, 3, 4, \ldots\}$ which is all the

natural numbers except $0$, and then compare it to the set

$\{0, 1, 2, 3, 4, \ldots\}$ which contains the $0$ and all other

numbers. If you think, the second must be more infinite\ldots{} well,

then think again. Because the two infinite sets

\begin{center}

  $\{1, 2, 3, 4, \ldots\}$ and

  $\{0, 1, 2, 3, 4, \ldots\}$

\end{center}

\noindent

contain actually the same amount of elements. Does this make sense?

Though this might all look strange, infinite sets will be a

topic that is very relevant to the material of this module. It tells

us what we can compute with a computer (actually an algorithm) and what

we cannot. But during the first 9 lectures we can go by without this

``weird'' stuff. End of aside.\smallskip

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 at all! Make sure you see the difference.

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 The cool thing about Scala is that we can define language

concatenation very elegantly as

\begin{lstlisting}[numbers=none]

def concat(A: Set[String], B: Set[String]) =

  for (x <- A ; y <- B) yield x ++ y

\end{lstlisting}

\noindent

where \code{++} is string concatenation in Scala.

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, whereas the set $\{[]\}$

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

Again this is a subtletly you need to get compfortable with.

Using the operation of 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 in mathematics 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} after the mathematician/computer scientist

Stephen Cole Kleene. Unfolding the definition in~\eqref{star}

gives

\[

A\star \dn A^0 \cup A^1 \cup A^2 \cup A^3 \cup \ldots

\]

\noindent

which is equal to 

\[

A\star \dn \{[]\} \,\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 as $\Sigma\star$. In doing so we also include the

empty string as a possible string (over $\Sigma$). Assuming $\Sigma

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

\[

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

\]

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

$b$s only, plus the empty string.

\bigskip

\noindent

Thanks for making it until here! There are also some personal conventions

about regular expressions. But I will explain them in the handout for the

first week. An exercise you can do: Implement the power operation for languages

and try out some examples.

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