LECTURE 1
THEORY OF
AUTOMATA
PRAGMATICS

Pre-Requisites


Text book


Introduction to Computer Theory by Daniel I.A. Cohen
Quizzes


No Pre-Requisite
Roughly once every two weeks
Marks Distribution

Theoretical Course

20% Mid-term, 40% Final, 15 % Quizzes, 25 %
Assignments + project.
COURSE OBJECTIVES

The major objective of this course is to introduce the student to the
concepts of theory of automata in computer science. The student
should acquire insights into the relationship among formal
languages, formal grammars, and automata.

Upon successful completion of this course, students will be able to:

Understand the equivalence between Nondeterministic Finite
State Automata and Deterministic Finite State Automata.

Understand the equivalence between Context-Free Grammars
and Nondeterministic Pushdown Automata.

Appreciate the power of the Turing Machine, as an abstract
automaton, that describes computation, effectively and
efficiently.
WHAT DOES “AUTOMATA” MEAN?

It is the plural of automaton, and it means
“something that works automatically”
AUTOMATA THEORY
Deals with the definitions and properties of
mathematical model of computation.
 Examples: Finite automata, Context free
grammars.
 Finite Automaton: Text Processing, Compilers
 Context Free grammars: Programming
languages, AI

SETS
A
set is a group of objects, called elements
(or members) of this set. For example, the
students in this room form a set.
 A set can be defined by listing all its
elements inside braces, e.g.:

S ={ 7,21,57}
 The
order of elements in sets do not
matter – in particular, S={7,21,57} = {21,57,7}
SETS CONTINUED…
 The
membership is denoted by ϵ symbol.
For example, 21 ϵ S but 10 not belong to S.
 If every member of A is also a member of
B. We say that A is a subset of B and if A
is a subset of B but not equal to B then A
is proper subset of B and write as
A  B. If A is a subset of B and A is equal
to B then A is improper subset of B and
write as A  B.
EXAMPLES OF SETS
 The
set with no elements is called the
empty set and denoted by Λ
 The empty set is a subset of every set.
 The set of natural numbers N (or N):
 N = {1, 2, 3, . . .}
 The set of integers Z (or Z):

Z = {. . ., -2,-1, 0, 1, 2,…}
 It
is clear that N subset of Z
SET OPERATIONS
LANGUAGES


In English, we distinguish 3 different entities:
letters, words, and sentences.

Groups of letters make up words and groups of
words make up sentences.

However, not all collections of letters form valid
words, and not all collections of words form valid
sentences.
This situation also exists with computer
languages.

Certain (but not all) strings of characters are
recognizable words (e.g., IF, ELSE, FOR, WHILE
…).
BASIC DEFINITIONS
A finite non-empty set of symbols (letters), is
called an alphabet.
 It is denoted by Σ ( Greek letter sigma).

Example:
Σ={a,b}
Σ={0,1} //important as this is the language
//which the computer understands.
Σ={i,j,k}
STRINGS
Concatenation of finite symbols from the alphabet
is called a string.

Example:
If Σ= {a,b} then
a, abab, aaabb, ababababababababab
WORDS

Words are strings belonging to some language.
Example:
If Σ= {x} then a language L can be defined as
L={xn : n=1,2,3,…..} or L={x,xx,xxx,….}
Here x,xx,… are the words of L
EMPTY STRING OR NULL STRING
We shall allow a string to have no letters. We call
this empty string or null string, and denote it
by the symbol Λ.
 For all languages, the null word, if it is a word
in the language, is the word that has no letters.
We also denote the null word by Λ.

STRING OPERATIONS
w  a1a2 an
abba
v  b1b2 bm
bbbaaa
Concatenation
wv  a1a2 anb1b2 bm
abbabbbaaa
w  a1a2 an
ababaaabbb
Reverse
w  an a2a1
R
bbbaaababa
STRING LENGTH
w  a1a2 an
w n
Length:
Examples:
abba  4
aa  2
a 1
DEFINING LANGUAGES


Example: Consider this alphabet with only one
letter
∑={x}
We can define a language by saying that any
nonempty string of alphabet letters is a
word
L1 = { x, xx, xxx, xxxx, … } or
L1 = { xn for n = 1, 2, 3, … }
Note that because of the way we have defined it,
the language L1 does not include the null word Λ.
EXAMPLE:
The language L of strings of odd length, defined
over Σ={a}, can be written as
L={a, aaa, aaaaa,…..}
 Example:
The language L of strings that does not start
with a, defined over Σ={a,b,c}, can be written as
L={b, c, ba, bb, bc, ca, cb, cc, …}
EXAMPLE:
The language L of strings of length 2, defined over
Σ={0,1,2}, can be written as
L={00, 01, 02,10, 11,12,20,21,22}

Example:
The language L of strings ending in 0, defined
over Σ ={0,1}, can be written as
L={0,00,10,000,010,100,110,…}
EXAMPLE:


The language EQUAL, of strings with number of
a’s equal to number of b’s, defined over Σ={a,b},
can be written as
{Λ ,ab,aabb,abab,baba,abba,…}
The language EVEN-EVEN, of strings with even
number of a’s and even number of b’s, defined
over Σ={a,b}, can be written as
{Λ, aa, bb, aaaa,aabb,abab, abba, baab, baba,
bbaa, bbbb,…}