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Theory of Automata

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Theory of Automata
(CS402)
Theory of Automata
Lecture N0. 6
Reading Material
Chapter 5
Introduction to Computer Theory
Summary
Language of strings beginning with and ending in different letters, Accepting all strings, accepting non-
empty strings, accepting no string, containing double a's, having double 0's or double 1's, containing
triple a's or triple b's, EVEN-EVEN
Example
Consider the Language L of Strings, defined over Σ = {a, b}, beginning with and ending in different letters.
The language L may be expressed by the following regular expression a(a + b)*b + b(a + b)*a
This language may be accepted by the following FA  a
b
b
4+
2
a
a
b
a
a
b
5+
3
b
Example
Consider the Language L, defined over Σ = {a, b} of all strings including Λ. The language L may be accepted
a,b
by the following FA
a,b
1±
2+
The language L may also be accepted by the following FA
a,b
±
The language L may be expressed by the regular expression (a + b)*
Example
Consider the Language L , defined over Σ = {a, b} of all non empty strings. The language L may be accepted
a,b
by the following FA
a,b
-
+
The above language may be expressed by the regular expression (a + b)+
Example
Consider the following FA, defined over Σ = {a, b}
a,b
a,b
-
+
It is to be noted that the above FA does not accept any string, even it does not accept the null string; as there is
no path starting from initial state and ending in final state.
Equivalent FAs
It is to be noted that two FAs are said to be equivalent, if they accept the same language, as shown in the
following FAs.
a,b
FA1
a,b
­
+
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Theory of Automata
(CS402)
a,b
a,b
1-
2
FA2
a,b
a,b
a,b
3+
1-
2
FA3
±
Note
FA1 has already been discussed, while in FA2, there is no final state and in FA3, there is a final state but FA3 is
disconnected as the states 2 and 3 are disconnected.
It may also be noted that the language of strings accepted by FA1, FA2 and FA3 is denoted by the empty set i.e.
{ } OR Ř
Example
Consider the Language L of strings , defined over Σ = {a, b}, containing double a.
The language L may be expressed by the regular expression (a+b)* (aa) (a+b)*. This language may be accepted
by the following FA.
b
a,b
a
a
2
1-
3+
b
Example
Consider the language L of strings, defined over Σ={0, 1}, having double 0's or double 1's, The language L
may be expressed by the regular expression (0+1)* (00 + 11) (0+1)*
This language may be accepted by the following FA
x
0
0
0,1
1
0
+
-
1
1
y
18
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Theory of Automata
(CS402)
Example
Consider the language L of strings, defined over Σ={a, b}, having triple a's or triple b's. The language L may
be expressed by RE (a+b)* (aaa + bbb) (a+b)*
a
2
4
This language may be accepted by the FA as shown aside
a
a
a,b
b
6+
a
b
b
a
b
b
3
5
3
b
1±
3
b
a
a
a
a
b
4
2
b
Example
Consider the EVEN-EVEN language, defined over Σ = {a, b}. As discussed earlier that EVEN-EVEN
language can be expressed by the regular expression (aa+bb+(ab+ba)(aa+bb)*(ab+ba))*
EVEN-EVEN language may be accepted by the FA as shown aside
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