Regular Expression, Recursive definition of Regular Expression

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Equivalent Regular Expressions >>

Theory of Automata

(CS402)

Theory of Automata

Lecture N0. 3

Reading Material

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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