# Theory of Automata

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Theory of Automata
(CS402)
Theory of Automata
Lecture N0. 3
Chapter 4
Introduction to Computer Theory
Summary
RE, Recursive definition of RE, defining languages by RE, { x}*, { x}+, {a+b}*, Language of strings having
exactly one a, Language of strings of even length, Language of strings of odd length, RE defines unique
language (as Remark), Language of strings having at least one a, Language of strings having at least one a and
one b, Language of strings starting with aa and ending in bb, Language of strings starting with and ending in
different letters.
Regular Expression
As discussed earlier that a* generates Λ, a, aa, aaa, ... and a+ generates a, aa, aaa, aaaa, ..., so the language L1
= {Λ, a, aa, aaa, ...} and L2 = {a, aa, aaa, aaaa, ...} can simply be expressed by a* and a+, respectively.
a* and a+ are called the regular expressions (RE) for L1 and L2 respectively.
a+, aa* and a*a generate L2.
Note
Recursive definition of Regular Expression(RE)
Step 1: Every letter of Σ including Λ is a regular expression.
Step 2: If r1 and r2 are regular expressions then
(r1)
r1 r2
r1 + r2 and
r1*
are also regular expressions.
Step 3: Nothing else is a regular expression.
Method 3 (Regular Expressions)
Consider the language L={Λ, x, xx, xxx,...} of strings, defined over Σ = {x}.
We can write this language as the Kleene star closure of alphabet Σ or L=Σ*={x}* .
This language can also be expressed by the regular expression x*.
Similarly the language L={x, xx, xxx,...}, defined over Σ = {x}, can be expressed by the regular expression x+.
Now consider another language L, consisting of all possible strings, defined over Σ = {a, b}. This language can
also be expressed by the regular expression (a + b)*.
Now consider another language L, of strings having exactly one a, defined over Σ = {a, b}, then it's regular
expression may be b*ab*.
Now consider another language L, of even length, defined over Σ = {a, b}, then it's regular expression may be
((a+b)(a+b))*.
Now consider another language L, of odd length, defined over Σ = {a, b}, then it's regular expression may be
(a+b)((a+b)(a+b))* or ((a+b)(a+b))*(a+b).
Remark
It may be noted that a language may be expressed by more than one regular expression, while given a regular
expression there exist a unique language generated by that regular expression.
Example
Consider the language, defined over
Σ = {a , b} of words having at least one a, may be expressed by a regular expression (a+b)*a(a+b)*.
Consider the language, defined over Σ = {a, b} of words having at least one a and one b, may be expressed by a
regular expression (a+b)*a(a+b)*b(a+b)*+ (a+b)*b(a+b)*a(a+b)*.
Consider the language, defined over Σ ={a, b}, of words starting with double a and ending in double b then its
regular expression may be aa(a+b)*bb
Consider the language, defined over Σ ={a, b} of words starting with a and ending in b OR
starting with b and ending in a, then its regular expression may be a(a+b)*b+b(a+b)*a
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