# Theory of Automata

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(CS402)
Theory of Automata
Lecture N0. 2
Chapter 3
Introduction to Computer Theory
Summary
Kleene Star Closure, Plus operation, recursive definition of languages, INTEGER, EVEN, factorial,
PALINDROME, {anbn}, languages of strings (i) ending in a, (ii) beginning and ending in same letters, (iii)
containing aa or bb (iv) containing exactly one a
Kleene Star Closure
Given Σ, then the Kleene Star Closure of the alphabet Σ, denoted by Σ*, is the collection of all strings defined
over Σ, including Λ.
It is to be noted that Kleene Star Closure can be defined over any set of strings.
Examples
If Σ = {x}
Then Σ* = {Λ, x, xx, xxx, xxxx, ....}
If Σ = {0,1}
Then Σ* = {Λ, 0, 1, 00, 01, 10, 11, ....}
If Σ = {aaB, c}
Then Σ* = {Λ, aaB, c, aaBaaB, aaBc, caaB, cc, ....}
Note
Languages generated by Kleene Star Closure of set of strings, are infinite languages. (By infinite language, it is
supposed that the language contains infinite many words, each of finite length).
PLUS Operation (+)
Plus Operation is same as Kleene Star Closure except that it does not generate Λ (null string), automatically.
Example
If Σ = {0,1}
Then Σ+ = {0, 1, 00, 01, 10, 11, ....}
If Σ = {aab, c}
Then Σ+ = {aab, c, aabaab, aabc, caab, cc, ....}
Remark
It is to be noted that Kleene Star can also be operated on any string i.e. a* can be considered to be all possible
strings defined over {a}, which shows that a* generates Λ, a, aa, aaa, ...
It may also be noted that a+ can be considered to be all possible non empty strings defined over {a}, which
shows that a+ generates a, aa, aaa, aaaa, ...
Recursive definition of languages
The following three steps are used in recursive definition
Some basic words are specified in the language.
Rules for constructing more words are defined in the language.
No strings except those constructed in above, are allowed to be in the language.
Example
Defining language of INTEGER
Step 1:
1 is in INTEGER.
Step 2:
If x is in INTEGER then x+1 and x-1 are also in INTEGER.
Step 3:
No strings except those constructed in above, are allowed to be in INTEGER.
Example
9 Theory of Automata
(CS402)
Defining language of EVEN
Step 1:
2 is in EVEN.
Step 2:
If x is in EVEN then x+2 and x-2 are also in EVEN.
Step 3:
No strings except those constructed in above, are allowed to be in EVEN.
Example
Defining the language factorial
Step 1:
As 0!=1, so 1 is in factorial.
Step 2:
n!=n*(n-1)! is in factorial.
Step 3:
No strings except those constructed in above, are allowed to be in factorial.
Defining the language PALINDROME, defined over Σ = {a,b}
a and b are in PALINDROME
Step 1:
if x is palindrome, then s(x)Rev(s) and xx will also be palindrome, where s belongs to Σ*
Step 2:
Step 3:
No strings except those constructed in above, are allowed to be in palindrome
Defining the language {anbn }, n=1,2,3,... , of strings defined over Σ={a,b}
ab is in {anbn}
Step 1:
if x is in {anbn}, then axb is in {anbn}
Step 2:
No strings except those constructed in above, are allowed to be in {anbn}
Step 3:
Defining the language L, of strings ending in a , defined over Σ={a,b}
Step 1:
a is in L
if x is in L then s(x) is also in L, where s belongs to Σ*
Step 2:
Step 3:
No strings except those constructed in above, are allowed to be in L
Defining the language L, of strings beginning and ending in same letters , defined over Σ={a, b}
Step 1:
a and b are in L
(a)s(a) and (b)s(b) are also in L, where s belongs to Σ*
Step 2:
Step 3:
No strings except those constructed in above, are allowed to be in L
Defining the language L, of strings containing aa or bb , defined over
Σ={a, b}
Step 1:
aa and bb are in L
s(aa)s and s(bb)s are also in L, where s belongs to Σ*
Step 2:
No strings except those constructed in above, are allowed to be in L
Step 3:
Defining the language L, of strings containing exactly one a, defined over
Σ={a, b}
Step 1:
a is in L
s(aa)s is also in L, where s belongs to b*
Step 2:
Step 3:
No strings except those constructed in above, are allowed to be in L
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