# Theory of Automata

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(CS402)
Theory of Automata
Lecture N0. 9
Chapter 6
Introduction to Computer Theory
Summary
TGs accepting the languages: containing aaa or bbb, beginning and ending in different letters,
beginning and ending in same letters, EVEN-EVEN, a's occur in even clumps and ends in
three or more b's, example showing different paths traced by one string, Definition of GTG
Example
Consider the language L of strings, defined over Σ = {a, b}, having triple a or triple b. The language L may
be expressed by RE (a+b)* (aaa + bbb) (a+b)*
This language may be accepted by the following TG
a
2
4
a
a
a,b
a,b
6+
b
b
b
3
5
OR
OR
Example
Consider the language L of strings, defined over Σ = {a, b}, beginning and ending in different letters.
The language L may be expressed by RE a(a + b)*b + b(a + b)*a
The language L may be accepted by the following TG
25 Theory of Automata
(CS402)
Example
Consider the Language L of strings of length two or more, defined over Σ = {a, b}, beginning with and
ending in same letters.
The language L may be expressed by the following regular expression a(a + b)*a + b(a + b)*b
This language may be accepted by the following TG
Example
Consider the EVEN-EVEN language, defined over Σ = {a, b}. As discussed earlier that EVEN-EVEN
language can be expressed by a regular expression  (aa+bb+(ab+ba)(aa+bb)*(ab+ba))*
The language EVEN-EVEN may be accepted by the following TG
Example
Consider the language L, defined over Σ={a, b}, in which a's occur only in even clumps and that ends in
three or more b's. The language L can be expressed by its regular expression  (aa)*b(b*+(aa(aa)*b)*) bb OR
(aa)*b(b*+( (aa)+b)*) bb.
The language L may be accepted by the following TG
Example
Consider the following TG
26 Theory of Automata
(CS402)
Consider the string abbbabbbabba. It may be observed that the above string traces the following three paths,
(using the states)
(a)(b) (b) (b) (ab) (bb) (a) (bb) (a)
(-)(4)(4)(+)(+)(3)(2)(2)(1)(+)
(a)(b) ((b)(b)) (ab) (bb) (a) (bb) (a)
(-)(4)(+)(+)(+)(3)(2)(2)(1)(+)
(a)((b) (b)) (b) (ab) (bb) (a) (bb) (a)
(-) (4)(4)(4)(+) (3)(2)(2)(1)(+)
Which shows that all these paths are successful, (i.e. the path starting from an initial state and ending in a final
state).
Hence the string abbbabbbabba is accepted by the given TG.
Generalized Transition Graphs
A generalized transition graph (GTG) is a collection of three things
Finite number of states, at least one of which is start state and some (maybe none) final states.
Finite set of input letters (Σ) from which input strings are formed.
Directed edges connecting some pair of states labeled with regular expression.
It may be noted that in GTG, the labels of transition edges are corresponding regular expressions
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