Theory of Automata

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Welcome to !
Theory Of Automata

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Text and Reference Material
1. Introduction to Computer Theory, by Daniel I.
Cohen, John Wiley and Sons, Inc., 1991,
Second Edition
2. Introduction to Languages and Theory of
Computation, by J. C. Martin, McGraw Hill
Book Co., 1997, Second Edition

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Grading
There will be One term exam and one final
exam. The final exam will be comprehensive.
These will contribute the following percentages
to the final grade:
Mid-Term Exams. 35%
Assignments 15%
Final Exams. 50%

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What does automata mean?
It is the plural of automaton, and it means
“something that works automatically”

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Introduction to languages
There are two types of languages
Formal Languages (Syntactic languages)
Informal Languages (Semantic
languages)

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Alphabets
Definition:
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}

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NOTE:
A certain version of language ALGOL has
113 letters
Σ (alphabet) includes letters, digits and a
variety of operators including sequential
operators such as GOTO and IF

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Strings
Definition:
Concatenation of finite symbols from the
alphabet is called a string.
Example:
If Σ= {a,b} then
a, abab, aaabb, ababababababababab

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NOTE:
EMPTY STRING or NULL STRING
Sometimes a string with no symbol at all is
used, denoted by (Small Greek letter Lambda)
λ or (Capital Greek letter Lambda) Λ, is called
an empty string or null string.
The capital lambda will mostly be used to
denote the empty string, in further discussion.

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Words
Definition:
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

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NOTE:
All words are strings, but not all strings
are words.

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Valid/In-valid alphabets
While defining an alphabet, an alphabet may
contain letters consisting of group of symbols
for example Σ1= {B, aB, bab, d}.
Now consider an alphabet
Σ2= {B, Ba, bab, d} and a string BababB.

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This string can be tokenized in two different
ways
(Ba), (bab), (B)
(B), (abab), (B)
Which shows that the second group cannot
be identified as a string, defined over
Σ = {a, b}.

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As when this string is scanned by the
compiler (Lexical Analyzer), first symbol B is
identified as a letter belonging to Σ, while for
the second letter the lexical analyzer would
not be able to identify, so while defining an
alphabet it should be kept in mind that
ambiguity should not be created.

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Remarks:
While defining an alphabet of letters
consisting of more than one symbols, no
letter should be started with the letter of the
same alphabet i.e. one letter should not be
the prefix of another.
However, a letter may be ended in the letter
of same alphabet i.e. one letter may be the
suffix of another.

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Conclusion
Σ1= {B, aB, bab, d}
Σ2= {B, Ba, bab, d}
Σ1 is a valid alphabet while Σ2 is an in-valid
alphabet.

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Length of Strings
Definition:
The length of string s, denoted by |s|, is the
number of letters in the string.
Example:
Σ={a,b}
s=ababa
|s|=5

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Example:
Σ= {B, aB, bab, d}
s=BaBbabBd
Tokenizing=(B), (aB), (bab), (d)
|s|=4

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Reverse of a String
Definition:
The reverse of a string s denoted by Rev(s)
or sr
, is obtained by writing the letters of s in
reverse order.
Example:
If s=abc is a string defined over Σ={a,b,c}
then Rev(s) or sr
= cba

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Example:
Σ= {B, aB, bab, d}
s=BaBbabBd
Rev(s)=dBbabaBB

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Lecture 2
Defining Languages
The languages can be defined in different
ways , such as Descriptive definition,
Recursive definition, using Regular
Expressions(RE) and using Finite
Automaton(FA) etc.
Descriptive definition of language:
The language is defined, describing the
conditions imposed on its words.

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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, …}

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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,…}

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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,…}
Example: 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,…}

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Example: The language INTEGER, of strings
defined over Σ={-,0,1,2,3,4,5,6,7,8,9}, can be
written as
INTEGER = {…,-2,-1,0,1,2,…}
Example: The language EVEN, of stings
defined over Σ={-,0,1,2,3,4,5,6,7,8,9}, can be
written as
EVEN = { …,-4,-2,0,2,4,…}

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Example: The language {an
bn
}, of strings defined
over Σ={a,b}, as
{an
bn
: n=1,2,3,…}, can be written as
{ab, aabb, aaabbb,aaaabbbb,…}
Example: The language {an
bn
an
}, of strings
defined over Σ={a,b}, as
{an
bn
an
{aba, aabbaa, aaabbbaaa,aaaabbbbaaaa,…}

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Example: The language factorial, of strings
defined over Σ={1,2,3,4,5,6,7,8,9} i.e.
{1,2,6,24,120,…}
Example: The language FACTORIAL, of
strings defined over Σ={a}, as
{an!
{a,aa,aaaaaa,…}. It is to be noted that the
language FACTORIAL can be defined over
any single letter alphabet.

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Example: The language
DOUBLEFACTORIAL, of strings defined over
Σ={a, b}, as
{an!
bn!
{ab, aabb, aaaaaabbbbbb,…}
Example: The language SQUARE, of strings
defined over Σ={a}, as
{an2
{a, aaaa, aaaaaaaaa,…}

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Example: The language
DOUBLESQUARE, of strings defined
over Σ={a,b}, as
{an2
bn2
{ab, aaaabbbb, aaaaaaaaabbbbbbbbb,…}

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Example: The language PRIME, of
strings defined over Σ={a}, as
{ap
: p is prime}, can be written as
{aa,aaa,aaaaa,aaaaaaa,aaaaaaaaaaa…}

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An Important language
PALINDROME:
The language consisting of Λ and the
strings s defined over Σ such that Rev(s)=s.
It is to be denoted that the words of
PALINDROME are called palindromes.
Example:For Σ={a,b},
PALINDROME={Λ , a, b, aa, bb, aaa, aba,
bab, bbb, ...}

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Remark
There are as many palindromes of length 2n
as there are of length 2n-1.
To prove the above remark, the following is
to be noted:

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Note
Number of strings of length ‘m’ defined over
alphabet of ‘n’ letters is nm
.
Examples:
The language of strings of length 2, defined
over Σ={a,b} is L={aa, ab, ba, bb} i.e. number
of strings = 22
The language of strings of length 3, defined
over Σ={a,b} is L={aaa, aab, aba, baa, abb,
bab, bba, bbb} i.e. number of strings = 23

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To calculate the number of palindromes of
length(2n), consider the following diagram,

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which shows that there are as many
palindromes of length 2n as there are the strings
of length n i.e. the required number of
palindromes are 2n
.

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To calculate the number of palindromes of
length (2n-1) with ‘a’ as the middle letter,
consider the following diagram,

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which shows that there are as many
palindromes of length 2n-1 as there are the
strings of length n-1 i.e. the required number of
palindromes are 2n-1
.
Similarly the number of palindromes of length
2n-1, with ‘ b ’ as middle letter, will be 2n-1
as well.
Hence the total number of palindromes of length
2n-1 will be 2n-1
+ 2n-1
= 2 (2n-1
)= 2n
.

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Exercise
Q) Prove that there are as many palindromes
of length 2n, defined over Σ = {a,b,c}, as
there are of length 2n-1. Determine the
number of palindromes of length 2n defined
over the same alphabet as well.

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SummingUp Lecture-1
Introduction to the course title, Formal and In-
formal languages, Alphabets, Strings, Null string,
Words, Valid and In-valid alphabets, length of a
string, Reverse of a string, Defining languages,
Descriptive definition of languages, EQUAL,
EVEN-EVEN, INTEGER, EVEN, { an
bn
}, { an
bn
an
},
factorial, FACTORIAL, DOUBLEFACTORIAL,
SQUARE, DOUBLESQUARE, PRIME,
PALINDROME.

Theory of Automata

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Theory of Automata