13 Regular Expressions

Regular Expressions

This ppt is the work by Dr. Costas Busch, used with permission, and available from http://csc.lsu.edu/~busch/courses/theorycomp/fall2008/ 1

Regular Expressions
Regular expressions describe regular languages

Example:

(a  b c) * describes the language

 a, bc *   , a, bc, aa, abc, bca,...
2

Recursive Definition
Primitive regular expressions:  ,  , 
Given regular expressionsr1

r2 and r1  r2 r1 r2 r1 *

Are regular expressions

 r1 
3

Examples
A regular expression:

 a  b c  * (c   )

Not a regular expression:

 a  b 

4

Languages of Regular Expressions

L r 

r
: language of regular expression

Example

L (a  b c) *   , a, bc, aa, abc, bca,...

5

Definition
For primitive regular expressions:
L   
L     
L a   a

6

Definition (continued)
For regular expressionsr1

andr2

L r1  r2  L r1   L r2 
L r1 r2  L r1  L r2 
L r1 *  L r1   *
L  r1   L r1 
7

Example
Regular expression: a  b  a *

L  a  b  a * L  a  b   L a *
L a  b  L a *
 L a   L b    L a   *
  a   b    a  *
 a, b   , a, aa, aaa,...

 a, aa, aaa,..., b, ba, baa,...
8

Example
Regular expression

r  a  b  *  a  bb 

L r   a, bb, aa, abb, ba, bbb,...

9

Example
Regular expression

r  aa  *  bb  * b

2n 2m

L r  {a b

b : n, m 0}

10

Example
Regular expression

r (0  1) * 00 (0  1) *

L(r ) = { all strings containing substring 00

11

Example
Regular expression r

(1  01) * (0   )

L(r ) = { all strings without substring 00 }

12

Equivalent Regular Expressions
Definition:
Regular expressions r1

andr2

are equivalent ifL( r1 ) L( r2 )

13

Example

L

= { all strings without substring 00 }

r1 (1  01) * (0   ) r2 (1* 011*) * (0   )  1* (0   )
L(r1 ) L(r2 ) L

r1

and r2 are equivalent regular expressions
14

Regular Expressions and Regular Languages

15

Theorem
Languages
Generated by